SlideShare a Scribd company logo

Solving Inequalities Notes

L
Lori Rapp

The document describes how to solve an inequality problem step-by-step: (1) isolate the variable on one side of the inequality sign using the opposite operations of addition, subtraction, multiplication, and division; (2) determine whether the coefficient of the variable is 1; and (3) check the solution by graphing the inequality. It provides an example problem of solving 2x - 7 > 11 to illustrate the process.

EducationTechnologyEconomy & Finance
1 of 193
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
Solving Inequalities
The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks?
The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Think about how to solve this problem.
The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Think about how to solve this problem. • What number did you come up with?
The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Think about how to solve this problem. • What number did you come up with? • Ask yourself if the number makes sense in the context of this problem.
The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Think about how to solve this problem. • What number did you come up with? • Ask yourself if the number makes sense in the context of this problem. • How did you solve this problem?
The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Think about how to solve this problem. • What number did you come up with? • Ask yourself if the number makes sense in the context of this problem. • How did you solve this problem? • Keep your answers and we’ll return to this problem later.
Recall Solving Equations 2x − 7 = 11
Recall Solving Equations • Use the properties of equality to 2x − 7 = 11 isolate the variable.
Recall Solving Equations • Use the properties of equality to 2x − 7 = 11 isolate the variable. • Undo subtraction (or addition) first using the opposite operation.
Recall Solving Equations • Use the properties of equality to 2x − 7 = 11 isolate the variable. +7 +7 • Undo subtraction (or addition) first using the opposite operation.
Recall Solving Equations • Use the properties of equality to 2x − 7 = 11 isolate the variable. +7 +7 • Undo subtraction (or addition) first using the opposite operation. 2x = 18
Recall Solving Equations • Use the properties of equality to 2x − 7 = 11 isolate the variable. +7 +7 • Undo subtraction (or addition) first using the opposite operation. 2x = 18 • Then undo multiplication (or division) using the opposite operation.
Recall Solving Equations • Use the properties of equality to 2x − 7 = 11 isolate the variable. +7 +7 • Undo subtraction (or addition) first using the opposite operation. 2x = 18 • Then undo multiplication (or division) 2 2 using the opposite operation.
Recall Solving Equations • Use the properties of equality to 2x − 7 = 11 isolate the variable. +7 +7 • Undo subtraction (or addition) first using the opposite operation. 2x = 18 • Then undo multiplication (or division) 2 2 using the opposite operation. x=9
Recall Solving Equations • Use the properties of equality to 2x − 7 = 11 isolate the variable. +7 +7 • Undo subtraction (or addition) first using the opposite operation. 2x = 18 • Then undo multiplication (or division) 2 2 using the opposite operation. x=9 • The coefficient of x is 1. Check your solution in the original problem.
Recall Solving Equations • Use the properties of equality to 2x − 7 = 11 isolate the variable. +7 +7 • Undo subtraction (or addition) first using the opposite operation. 2x = 18 • Then undo multiplication (or division) 2 2 using the opposite operation. x=9 • The coefficient of x is 1. Check your solution in the original problem. Check: ? 2 ⋅ 9 − 7 = 11 11 = 11
Recall Solving Equations • Use the properties of equality to 2x − 7 = 11 isolate the variable. +7 +7 • Undo subtraction (or addition) first using the opposite operation. 2x = 18 • Then undo multiplication (or division) 2 2 using the opposite operation. x=9 • The coefficient of x is 1. Check your solution in the original problem. Check: ? • It works so your solution is correct. 2 ⋅ 9 − 7 = 11 11 = 11
Solving Inequality - Example 1 2x − 7 > 11
Solving Inequality - Example 1 • Use the properties of equality to isolate the variable. 2x − 7 > 11
Solving Inequality - Example 1 • Use the properties of equality to isolate the variable. 2x − 7 > 11 • Undo subtraction (or addition) first using the opposite operation.
Solving Inequality - Example 1 • Use the properties of equality to isolate the variable. 2x − 7 > 11 • Undo subtraction (or addition) first using the +7 +7 opposite operation.
Solving Inequality - Example 1 • Use the properties of equality to isolate the variable. 2x − 7 > 11 • Undo subtraction (or addition) first using the +7 +7 opposite operation. 2x > 18
Solving Inequality - Example 1 • Use the properties of equality to isolate the variable. 2x − 7 > 11 • Undo subtraction (or addition) first using the +7 +7 opposite operation. 2x > 18 • Then undo multiplication (or division) using the opposite operation.
Solving Inequality - Example 1 • Use the properties of equality to isolate the variable. 2x − 7 > 11 • Undo subtraction (or addition) first using the +7 +7 opposite operation. 2x > 18 • Then undo multiplication (or division) using 2 2 the opposite operation.
Solving Inequality - Example 1 • Use the properties of equality to isolate the variable. 2x − 7 > 11 • Undo subtraction (or addition) first using the +7 +7 opposite operation. 2x > 18 • Then undo multiplication (or division) using 2 2 the opposite operation. x>9
Solving Inequality - Example 1 • Use the properties of equality to isolate the variable. 2x − 7 > 11 • Undo subtraction (or addition) first using the +7 +7 opposite operation. 2x > 18 • Then undo multiplication (or division) using 2 2 the opposite operation. • The coefficient of x is 1. x>9
Solving Inequality - Example 1 • Use the properties of equality to isolate the variable. 2x − 7 > 11 • Undo subtraction (or addition) first using the +7 +7 opposite operation. 2x > 18 • Then undo multiplication (or division) using 2 2 the opposite operation. • The coefficient of x is 1. x>9 • Steps were the same!
Solving Inequality - Example 1 • Use the properties of equality to isolate the variable. 2x − 7 > 11 • Undo subtraction (or addition) first using the +7 +7 opposite operation. 2x > 18 • Then undo multiplication (or division) using 2 2 the opposite operation. • The coefficient of x is 1. x>9 • Steps were the same! • Final step is to check the solution.
Example 1 (con’t) • Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10
Example 1 (con’t) • Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10 • Open circle at 9 because >, not ≥.
Example 1 (con’t) • Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10 • Open circle at 9 because >, not ≥. • Always shade in the direction of a number that works in the inequality.
Example 1 (con’t) • Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10 • Open circle at 9 because >, not ≥. • Always shade in the direction of a number that works in the inequality. • Shade to the right because 10 > 9.
Example 1 (con’t) • Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10 • Open circle at 9 because >, not ≥. • Always shade in the direction of a number that works in the inequality. • Shade to the right because 10 > 9. • Check the solution in the ORIGINAL problem.
Example 1 (con’t) • Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10 • Open circle at 9 because >, not ≥. • Always shade in the direction of a number that works in the inequality. Original Problem: • Shade to the right because 10 > 9. 2x − 7 > 11 • Check the solution in the ORIGINAL problem.
Example 1 (con’t) • Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10 • Open circle at 9 because >, not ≥. • Always shade in the direction of a number that works in the inequality. Original Problem: • Shade to the right because 10 > 9. 2x − 7 > 11 • Check the solution in the ORIGINAL problem. • Check the solution 10.
Example 1 (con’t) • Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10 • Open circle at 9 because >, not ≥. • Always shade in the direction of a number that works in the inequality. Original Problem: • Shade to the right because 10 > 9. 2x − 7 > 11 • Check the solution in the ORIGINAL Check: problem. ? 2 ⋅10 − 7 > 11 • Check the solution 10. 13 > 11
Example 1 (con’t) • Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10 • Open circle at 9 because >, not ≥. • Always shade in the direction of a number that works in the inequality. Original Problem: • Shade to the right because 10 > 9. 2x − 7 > 11 • Check the solution in the ORIGINAL Check: problem. ? 2 ⋅10 − 7 > 11 • Check the solution 10. 13 > 11 • This is true and therefore a solution.
Example 1 (con’t) • It is good to check at least 2 other numbers x>9 for inequalities. 0 1 2 3 4 5 6 7 8 9 10 Original Problem: 2x − 7 > 11 Check:
Example 1 (con’t) • It is good to check at least 2 other numbers x>9 for inequalities. 0 1 2 3 4 5 6 7 8 9 10 • Pick a number that is not a solution and the number where the circle is placed. Original Problem: 2x − 7 > 11 Check:
Example 1 (con’t) • It is good to check at least 2 other numbers x>9 for inequalities. 0 1 2 3 4 5 6 7 8 9 10 • Pick a number that is not a solution and the number where the circle is placed. Original Problem: • 0 is not a solution. 2x − 7 > 11 Check:
Example 1 (con’t) • It is good to check at least 2 other numbers x>9 for inequalities. 0 1 2 3 4 5 6 7 8 9 10 • Pick a number that is not a solution and the number where the circle is placed. Original Problem: • 0 is not a solution. 2x − 7 > 11 • The check shows -7 > 11, which is not true. This is good because it is not shaded. Check: ? 2 ⋅ 0 − 7 > 11 −7 > 11 x
Example 1 (con’t) • It is good to check at least 2 other numbers x>9 for inequalities. 0 1 2 3 4 5 6 7 8 9 10 • Pick a number that is not a solution and the number where the circle is placed. Original Problem: • 0 is not a solution. 2x − 7 > 11 • The check shows -7 > 11, which is not true. This is good because it is not shaded. Check: ? • 9 is not a solution because of the open 2 ⋅ 0 − 7 > 11 circle. −7 > 11 x
Example 1 (con’t) • It is good to check at least 2 other numbers x>9 for inequalities. 0 1 2 3 4 5 6 7 8 9 10 • Pick a number that is not a solution and the number where the circle is placed. Original Problem: • 0 is not a solution. 2x − 7 > 11 • The check shows -7 > 11, which is not true. This is good because it is not shaded. Check: ? • 9 is not a solution because of the open 2 ⋅ 0 − 7 > 11 circle. −7 > 11 x ? • 11 is not greater than 11. Again this is 2 ⋅ 9 − 7 > 11 good because 9 is not a solution. 11 > 11 x
Solving Inequality Recap
Solving Inequality Recap 1) Solve the inequality as you would an equation.
Solving Inequality Recap 1) Solve the inequality as you would an equation. 2) Graph the solution on a number line.
Solving Inequality Recap 1) Solve the inequality as you would an equation. 2) Graph the solution on a number line. 3) Check at least 3 numbers in the original inequality.
Solving Inequality Recap 1) Solve the inequality as you would an equation. 2) Graph the solution on a number line. 3) Check at least 3 numbers in the original inequality. 4) Write the solution.
Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. • Check solution in original inequality. • -5 is NOT greater than -3! • Why didn’t it work? • Negative means reverse. When dividing by a negative, don’t we reverse all the signs? • The same goes for the inequality. When dividing (or multiplying) by a negative, the inequality must reverse.
Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. −1 −1 • Check solution in original inequality. • -5 is NOT greater than -3! • Why didn’t it work? • Negative means reverse. When dividing by a negative, don’t we reverse all the signs? • The same goes for the inequality. When dividing (or multiplying) by a negative, the inequality must reverse.
Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. −1 −1 • Check solution in original inequality. • -5 is NOT greater than -3! −2x > −4 • Why didn’t it work? • Negative means reverse. When dividing by a negative, don’t we reverse all the signs? • The same goes for the inequality. When dividing (or multiplying) by a negative, the inequality must reverse.
Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. −1 −1 • Check solution in original inequality. • -5 is NOT greater than -3! −2x > −4 • Why didn’t it work? -2 -2 • Negative means reverse. When dividing by a negative, don’t we reverse all the signs? • The same goes for the inequality. When dividing (or multiplying) by a negative, the inequality must reverse.
Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. −1 −1 • Check solution in original inequality. • -5 is NOT greater than -3! −2x > −4 • Why didn’t it work? -2 -2 • Negative means reverse. When dividing x>2 by a negative, don’t we reverse all the signs? • The same goes for the inequality. When dividing (or multiplying) by a negative, the inequality must reverse.
Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. −1 −1 −2x > −4 -2 -2 x>2 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. −1 −1 −2x > −4 -2 -2 x>2 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. −1 −1 • Check solution in original inequality. −2x > −4 -2 -2 x>2 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. −1 −1 • Check solution in original inequality. −2x > −4 -2 -2 x>2 -5 -4 -3 -2 -1 0 1 2 3 4 5 Check: ? −2 ⋅ 3 + 1>− 3 −5 > −3
Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. −1 −1 • Check solution in original inequality. • -5 is NOT greater than -3! −2x > −4 -2 -2 x>2 -5 -4 -3 -2 -1 0 1 2 3 4 5 Check: ? −2 ⋅ 3 + 1>− 3 −5 > −3 x
Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. −1 −1 • Check solution in original inequality. • -5 is NOT greater than -3! −2x > −4 • Why didn’t it work? -2 -2 x>2 -5 -4 -3 -2 -1 0 1 2 3 4 5 Check: ? −2 ⋅ 3 + 1>− 3 −5 > −3 x
Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. −1 −1 • Check solution in original inequality. • -5 is NOT greater than -3! −2x > −4 • Why didn’t it work? -2 -2 • Negative means reverse. When dividing x>2 by a negative, don’t we reverse all the -5 -4 -3 -2 -1 0 1 2 3 4 5 signs? Check: ? −2 ⋅ 3 + 1>− 3 −5 > −3 x
Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. −1 −1 • Check solution in original inequality. • -5 is NOT greater than -3! −2x > −4 • Why didn’t it work? -2 -2 • Negative means reverse. When dividing x>2 by a negative, don’t we reverse all the -5 -4 -3 -2 -1 0 1 2 3 4 5 signs? Check: ? • The same goes for the inequality. When −2 ⋅ 3 + 1>− 3 dividing (or multiplying) by a negative, the inequality must reverse. −5 > −3 x
Example 2 (con’t) • Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. • Graph. • Check solution in the original inequality. • 1 is greater than -3! • Write the solution.
Example 2 (con’t) • Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1 • Graph. • Check solution in the original inequality. • 1 is greater than -3! • Write the solution.
Example 2 (con’t) • Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1 • Graph. −2x > −4 • Check solution in the original inequality. • 1 is greater than -3! • Write the solution.
Example 2 (con’t) • Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1 • Graph. −2x > −4 -2 -2 • Check solution in the original inequality. • 1 is greater than -3! • Write the solution.
Example 2 (con’t) • Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1 • Graph. −2x > −4 -2 -2 • Check solution in the original x<2 inequality. • 1 is greater than -3! • Write the solution.
Example 2 (con’t) • Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1 • Graph. −2x > −4 -2 -2 x<2
Example 2 (con’t) • Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1 • Graph. −2x > −4 -2 -2 x<2 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 2 (con’t) • Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1 • Graph. −2x > −4 -2 -2 • Check solution in the original x<2 inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 2 (con’t) • Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1 • Graph. −2x > −4 -2 -2 • Check solution in the original x<2 inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 Check: ? −2 ⋅ 0 + 1>− 3 1 > −3
Example 2 (con’t) • Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1 • Graph. −2x > −4 -2 -2 • Check solution in the original x<2 inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 • 1 is greater than -3! Check: ? −2 ⋅ 0 + 1>− 3 1 > −3
Example 2 (con’t) • Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1 • Graph. −2x > −4 -2 -2 • Check solution in the original x<2 inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 • 1 is greater than -3! Check: ? • Write the solution. −2 ⋅ 0 + 1>− 3 1 > −3
Example 2 (con’t) • Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1 • Graph. −2x > −4 -2 -2 • Check solution in the original x<2 inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 • 1 is greater than -3! Check: ? • Write the solution. −2 ⋅ 0 + 1>− 3 {x | x < 2} 1 > −3
Equation vs Inequality
Equation vs Inequality • What are the similarities between solving an Equation and solving an Inequality?
Equation vs Inequality • What are the similarities between solving an Equation and solving an Inequality? ✓ Solve using the same process.
Equation vs Inequality • What are the similarities between solving an Equation and solving an Inequality? ✓ Solve using the same process. ✓ Always check solution.
Equation vs Inequality
Equation vs Inequality • What are the differences between solving an Equation and solving an Inequality?
Equation vs Inequality • What are the differences between solving an Equation and solving an Inequality? ✓ When multiplying or dividing by a negative, the inequality reverses.
Equation vs Inequality • What are the differences between solving an Equation and solving an Inequality? ✓ When multiplying or dividing by a negative, the inequality reverses. ‣ < changes to >
Equation vs Inequality • What are the differences between solving an Equation and solving an Inequality? ✓ When multiplying or dividing by a negative, the inequality reverses. ‣ < changes to > ‣ > changes to <
Equation vs Inequality • What are the differences between solving an Equation and solving an Inequality? ✓ When multiplying or dividing by a negative, the inequality reverses. ‣ < changes to > ‣ > changes to < ‣ ≤ changes to ≥
Equation vs Inequality • What are the differences between solving an Equation and solving an Inequality? ✓ When multiplying or dividing by a negative, the inequality reverses. ‣ < changes to > ‣ > changes to < ‣ ≤ changes to ≥ ‣ ≥ changes to ≤
Equation vs Inequality • What are the differences between solving an Equation and solving an Inequality? ✓ When multiplying or dividing by a negative, the inequality reverses. ‣ < changes to > ‣ > changes to < ‣ ≤ changes to ≥ ‣ ≥ changes to ≤ ✓ Check multiple numbers to include solutions and non- solutions.
Example 3 - You try... • Solve, graph, check, and write your solution to 8x − 11 ≤ 13
Example 3 - You try... • Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11
Example 3 - You try... • Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 8x ≤ 24
Example 3 - You try... • Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 8x ≤ 24 8 8
Example 3 - You try... • Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 8x ≤ 24 8 8 x≤3
Example 3 - You try... • Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 -5 -4 -3 -2 -1 0 1 2 3 4 5 8x ≤ 24 8 8 x≤3
Example 3 - You try... • Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 -5 -4 -3 -2 -1 0 1 2 3 4 5 8x ≤ 24 8 8 x≤3
Example 3 - You try... • Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 -5 -4 -3 -2 -1 0 1 2 3 4 5 8x ≤ 24 Check: 8 8 x≤3
Example 3 - You try... • Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 -5 -4 -3 -2 -1 0 1 2 3 4 5 8x ≤ 24 Check: ? 8 8 8 ⋅ 0 − 11≤ 13 −11 ≤ 13 x≤3
Example 3 - You try... • Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 -5 -4 -3 -2 -1 0 1 2 3 4 5 8x ≤ 24 Check: ? ? 8 8 8 ⋅ 0 − 11≤ 13 8 ⋅ 5 − 11≤ 13 x≤3 −11 ≤ 13 29 ≤ 13 x
Example 3 - You try... • Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 -5 -4 -3 -2 -1 0 1 2 3 4 5 8x ≤ 24 Check: ? ? 8 8 8 ⋅ 0 − 11≤ 13 8 ⋅ 5 − 11≤ 13 x≤3 −11 ≤ 13 ? 29 ≤ 13 x 8 ⋅ 3 − 11≤ 13 13 ≤ 13
Example 3 - You try... • Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 -5 -4 -3 -2 -1 0 1 2 3 4 5 8x ≤ 24 Check: ? ? 8 8 8 ⋅ 0 − 11≤ 13 8 ⋅ 5 − 11≤ 13 x≤3 −11 ≤ 13 ? 29 ≤ 13 x 8 ⋅ 3 − 11≤ 13 13 ≤ 13 {x | x ≤ 3}
Example 4 - You try... • Solve, graph, check, and write your solution to −3x + 10 > −5
Example 4 - You try... • Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10
Example 4 - You try... • Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 −3x > −15
Example 4 - You try... • Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 −3x > −15 −3 −3
Example 4 - You try... • Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 −3x > −15 −3 −3 x<5
Example 4 - You try... • Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 -5 -4 -3 -2 -1 0 1 2 3 4 5 −3x > −15 −3 −3 x<5
Example 4 - You try... • Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 -5 -4 -3 -2 -1 0 1 2 3 4 5 −3x > −15 −3 −3 x<5
Example 4 - You try... • Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 -5 -4 -3 -2 -1 0 1 2 3 4 5 −3x > −15 Check: −3 −3 x<5
Example 4 - You try... • Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 -5 -4 -3 -2 -1 0 1 2 3 4 5 −3x > −15 Check: ? −3 −3 −3 ⋅ 0 + 10 >− 5 10 > −5 x<5
Example 4 - You try... • Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 -5 -4 -3 -2 -1 0 1 2 3 4 5 −3x > −15 Check: ? ? −3 −3 −3 ⋅ 0 + 10 >− 5 −3 ⋅10 + 10 >− 5 x<5 10 > −5 −20 > −5 x
Example 4 - You try... • Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 -5 -4 -3 -2 -1 0 1 2 3 4 5 −3x > −15 Check: ? ? −3 −3 −3 ⋅ 0 + 10 >− 5 −3 ⋅10 + 10 >− 5 x<5 10 > −5 −20 > −5 x ? −3 ⋅ 5 + 10 >− 5 −5 > −5 x
Example 4 - You try... • Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 -5 -4 -3 -2 -1 0 1 2 3 4 5 −3x > −15 Check: ? ? −3 −3 −3 ⋅ 0 + 10 >− 5 −3 ⋅10 + 10 >− 5 x<5 10 > −5 −20 > −5 x ? −3 ⋅ 5 + 10 >− 5 {x | x < 5} −5 > −5 x
Graphing Caution! • Did you notice that the shading always went in the direction of the inequality?
Graphing Caution! • Did you notice that the shading always went in the direction of the inequality? • Graph 1 > x
Graphing Caution! • Did you notice that the shading always went in the direction of the inequality? • Graph 1 > x -5 -4 -3 -2 -1 0 1 2 3 4 5
Graphing Caution! • Did you notice that the shading always went in the direction of the inequality? • Graph 1 > x -5 -4 -3 -2 -1 0 1 2 3 4 5 • Notice the shading is opposite the direction of the inequality. This is because x is on the right of the symbol instead of the left.
Graphing Caution! • Did you notice that the shading always went in the direction of the inequality? • Graph 1 > x -5 -4 -3 -2 -1 0 1 2 3 4 5 • Notice the shading is opposite the direction of the inequality. This is because x is on the right of the symbol instead of the left. • Always check numbers when determining shading!
What if parenthesis and/or variables on both sides of the inequality?
What if parenthesis and/or variables on both sides of the inequality? • Solve the same as you would if it was an equation.
What if parenthesis and/or variables on both sides of the inequality? • Solve the same as you would if it was an equation. • First, remove parenthesis.
What if parenthesis and/or variables on both sides of the inequality? • Solve the same as you would if it was an equation. • First, remove parenthesis. • Second, combine like terms on the same side of the inequality (if any).
What if parenthesis and/or variables on both sides of the inequality? • Solve the same as you would if it was an equation. • First, remove parenthesis. • Second, combine like terms on the same side of the inequality (if any). • Third, get variables on the same side of inequality.
What if parenthesis and/or variables on both sides of the inequality? • Solve the same as you would if it was an equation. • First, remove parenthesis. • Second, combine like terms on the same side of the inequality (if any). • Third, get variables on the same side of inequality. • Finally, solve for the variable.
Example 4 - 1− ( 4 + 3x ) < −3 − 5x
Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x
Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x 1− 4 − 3x < −3 − 5x
Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x • Combine like terms on the 1− 4 − 3x < −3 − 5x left.
Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x • Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x
Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x • Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x • Move variables on one side.
Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x • Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x • Move variables on one side. +5x +5x
Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x • Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x • Move variables on one side. +5x +5x • Solve for x.
Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x • Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x • Move variables on one side. +5x +5x • Solve for x. −3 + 2x < −3
Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x • Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x • Move variables on one side. +5x +5x • Solve for x. −3 + 2x < −3 +3 +3
Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x • Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x • Move variables on one side. +5x +5x • Solve for x. −3 + 2x < −3 +3 +3 2x < 0
Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x • Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x • Move variables on one side. +5x +5x • Solve for x. −3 + 2x < −3 +3 +3 2x < 0 2 2
Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x • Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x • Move variables on one side. +5x +5x • Solve for x. −3 + 2x < −3 +3 +3 2x < 0 2 2 x<0
Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x • Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x • Move variables on one side. +5x +5x • Solve for x. −3 + 2x < −3 +3 +3 • Graph solution. 2x < 0 2 2 x<0
Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x • Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x • Move variables on one side. +5x +5x • Solve for x. −3 + 2x < −3 +3 +3 • Graph solution. 2x < 0 2 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 x<0
Example 4 Continued... • Check at least 3 solutions 1− ( 4 + 3x ) < −3 − 5x in original inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 • The solution checks out.
Example 4 Continued... • Check at least 3 solutions 1− ( 4 + 3x ) < −3 − 5x in original inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 Check: • The solution checks out.
Example 4 Continued... • Check at least 3 solutions 1− ( 4 + 3x ) < −3 − 5x in original inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 ? Check: 1− ( 4 + 3⋅ −1) <− 3 − 5 ⋅ −1 1− 1 < −3 + 5 0<2 • The solution checks out.
Example 4 Continued... • Check at least 3 solutions 1− ( 4 + 3x ) < −3 − 5x in original inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 ? Check: ? 1− ( 4 + 3⋅ −1) <− 3 − 5 ⋅ −1 1− ( 4 + 3⋅1) <− 3 − 5 ⋅1 1− 1 < −3 + 5 1− 7 < −3 − 5 0<2 −6 < −8 x • The solution checks out.
Example 4 Continued... • Check at least 3 solutions 1− ( 4 + 3x ) < −3 − 5x in original inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 ? Check: ? 1− ( 4 + 3⋅ −1) <− 3 − 5 ⋅ −1 1− ( 4 + 3⋅1) <− 3 − 5 ⋅1 1− 1 < −3 + 5 1− 7 < −3 − 5 0<2 ? −6 < −8 x 1− ( 4 + 3⋅ 0 ) <− 3 − 5 ⋅ 0 −3 < −3 x • The solution checks out.
Example 4 Continued... • Check at least 3 solutions 1− ( 4 + 3x ) < −3 − 5x in original inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 ? Check: ? 1− ( 4 + 3⋅ −1) <− 3 − 5 ⋅ −1 1− ( 4 + 3⋅1) <− 3 − 5 ⋅1 1− 1 < −3 + 5 1− 7 < −3 − 5 0<2 ? −6 < −8 x 1− ( 4 + 3⋅ 0 ) <− 3 − 5 ⋅ 0 −3 < −3 x • The solution checks out.
Example 4 Continued... • Check at least 3 solutions 1− ( 4 + 3x ) < −3 − 5x in original inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 ? Check: ? 1− ( 4 + 3⋅ −1) <− 3 − 5 ⋅ −1 1− ( 4 + 3⋅1) <− 3 − 5 ⋅1 1− 1 < −3 + 5 1− 7 < −3 − 5 0<2 ? −6 < −8 x 1− ( 4 + 3⋅ 0 ) <− 3 − 5 ⋅ 0 −3 < −3 x • The solution checks out. {x | x < 0}
Example 5 - Solve 1 3(x − 5) ≥ ( 6x + 8 ) 2
Example 5 - Solve • Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 ) 2
Example 5 - Solve • Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 ) 2 3x − 15 ≥ 3x + 4
Example 5 - Solve • Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 ) • Move variables to one side. 2 3x − 15 ≥ 3x + 4
Example 5 - Solve • Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 ) • Move variables to one side. 2 3x − 15 ≥ 3x + 4 −3x −3x
Example 5 - Solve • Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 ) • Move variables to one side. 2 3x − 15 ≥ 3x + 4 −3x −3x −15 ≥ 4
Example 5 - Solve • Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 ) • Move variables to one side. 2 • What happen to x? 3x − 15 ≥ 3x + 4 −3x −3x −15 ≥ 4
Example 5 - Solve • Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 ) • Move variables to one side. 2 • What happen to x? 3x − 15 ≥ 3x + 4 • Is the resulting inequality −3x −3x true? −15 ≥ 4
Example 5 - Solve • Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 ) • Move variables to one side. 2 • What happen to x? 3x − 15 ≥ 3x + 4 • Is the resulting inequality −3x −3x true? −15 ≥ 4 • Because the result is a false statement, there is no solution to this inequality.
Example 5 - Solve • Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 ) • Move variables to one side. 2 • What happen to x? 3x − 15 ≥ 3x + 4 • Is the resulting inequality −3x −3x true? −15 ≥ 4 • Because the result is a false statement, there is no No Solution solution to this inequality.
Example 6 - Solve 2(5x + 4) ≥ 10x − 3
Example 6 - Solve • Remove parenthesis. 2(5x + 4) ≥ 10x − 3
Example 6 - Solve • Remove parenthesis. 2(5x + 4) ≥ 10x − 3 10x + 8 ≥ 10x − 3
Example 6 - Solve • Remove parenthesis. 2(5x + 4) ≥ 10x − 3 • Move variables to one side. 10x + 8 ≥ 10x − 3
Example 6 - Solve • Remove parenthesis. 2(5x + 4) ≥ 10x − 3 • Move variables to one side. 10x + 8 ≥ 10x − 3 −10x −10x
Example 6 - Solve • Remove parenthesis. 2(5x + 4) ≥ 10x − 3 • Move variables to one side. 10x + 8 ≥ 10x − 3 −10x −10x 8 ≥ −3
Example 6 - Solve • Remove parenthesis. 2(5x + 4) ≥ 10x − 3 • Move variables to one side. • The x disappeared again but is the 10x + 8 ≥ 10x − 3 resulting inequality true? −10x −10x 8 ≥ −3
Example 6 - Solve • Remove parenthesis. 2(5x + 4) ≥ 10x − 3 • Move variables to one side. • The x disappeared again but is the 10x + 8 ≥ 10x − 3 resulting inequality true? −10x −10x • Yes! Because the result is a true statement, any number can be 8 ≥ −3 substituted into the original inequality.
Example 6 - Solve • Remove parenthesis. 2(5x + 4) ≥ 10x − 3 • Move variables to one side. • The x disappeared again but is the 10x + 8 ≥ 10x − 3 resulting inequality true? −10x −10x • Yes! Because the result is a true statement, any number can be 8 ≥ −3 substituted into the original inequality. All Real Numbers • The graph is the entire number line so the solution is All Real -5 -4 -3 -2 -1 0 1 2 3 4 5 numbers.
Example 7 - The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks?
Example 7 - The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Remember this problem?
Example 7 - The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Remember this problem? • Would $2000 be enough?
Example 7 - The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Remember this problem? • Would $2000 be enough? • How about $710?
Example 7 - The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Remember this problem? • Would $2000 be enough? • How about $710? • What about $1710?
Example 7 - The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Remember this problem? • Would $2000 be enough? • How about $710? • What about $1710? • This situation can be modeled with an Inequality. We would use an Inequality instead of an Equation because an exact amount is not needed. There is a range of acceptable values.
(Example 7 - continued) The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Let A = Josh’s account balance before writing the checks.
(Example 7 - continued) The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Let A = Josh’s account balance before writing the checks. • The minimum balance is $1000 so this is the least amount allowed in the account.
(Example 7 - continued) The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Let A = Josh’s account balance before writing the checks. • The minimum balance is $1000 so this is the least amount allowed in the account. • Checks are deducted from the account balance. What operation shows a deduction?
(Example 7 - continued) The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Let A = Josh’s account balance before writing the checks. • The minimum balance is $1000 so this is the least amount allowed in the account. • Checks are deducted from the account balance. What operation shows a deduction? • Subtraction!
(Example 7 - continued) The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Let A = Josh’s account balance before writing the checks. • The minimum balance is $1000 so this is the least amount allowed in the account. • Checks are deducted from the account balance. What operation shows a deduction? • Subtraction! • $525 and $185 need subtracted from the account balance.
(Example 7 - continued) The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? Account balance - (sum of checks written) ≥ Min Balance
(Example 7 - continued) The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? Account balance - (sum of checks written) ≥ Min Balance A - (525 + 185) ≥ 1000
(Example 7 - continued) The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? Account balance - (sum of checks written) ≥ Min Balance A - (525 + 185) ≥ 1000 A - 710 ≥ 1000
(Example 7 - continued) The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? Account balance - (sum of checks written) ≥ Min Balance A - (525 + 185) ≥ 1000 A - 710 ≥ 1000 A - 710 + 710 ≥ 1000 + 710
(Example 7 - continued) The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? Account balance - (sum of checks written) ≥ Min Balance A - (525 + 185) ≥ 1000 A - 710 ≥ 1000 A - 710 + 710 ≥ 1000 + 710 A ≥ 1710
(Example 7 - continued) The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? Account balance - (sum of checks written) ≥ Min Balance A - (525 + 185) ≥ 1000 A - 710 ≥ 1000 A - 710 + 710 ≥ 1000 + 710 A ≥ 1710 Josh must have at least $1710 in his checking account before writing the checks.
Example 8 - Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover.
Example 8 - Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. • Let w = width
Example 8 - Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. • Let w = width • length = twice as long as the width = 2w.
Example 8 - Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. • Let w = width • length = twice as long as the width = 2w. • Enclosing a space refers to perimeter, the distance around the space.
Example 8 - Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. • Let w = width • length = twice as long as the width = 2w. • Enclosing a space refers to perimeter, the distance around the space. • This space is rectangular. Opposite sides have the same measure.
Example 8 - Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. • Let w = width • length = twice as long as the width = 2w. • Enclosing a space refers to perimeter, the distance around the space. • This space is rectangular. Opposite sides have the same measure. • Perimeter, P, of a rectangle is P = 2l + 2w, where l is length and w is width.
Example 8 - Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. • Let w = width • length = twice as long as the width = 2w. • Enclosing a space refers to perimeter, the distance around the space. • This space is rectangular. Opposite sides have the same measure. • Perimeter, P, of a rectangle is P = 2l + 2w, where l is length and w is width. • Because Jim has 96 feet of fence, this is the maximum perimeter.
(Example 8 - continued) Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. Maximum Perimeter ≥ 2l + 2w
(Example 8 - continued) Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. Maximum Perimeter ≥ 2l + 2w 96 ≥ 2 * 2w + 2 * w
(Example 8 - continued) Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. Maximum Perimeter ≥ 2l + 2w 96 ≥ 2 * 2w + 2 * w 96 ≥ 4w + 2w
(Example 8 - continued) Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. Maximum Perimeter ≥ 2l + 2w 96 ≥ 2 * 2w + 2 * w 96 ≥ 4w + 2w 96 ≥ 6w
(Example 8 - continued) Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. Maximum Perimeter ≥ 2l + 2w 96 ≥ 2 * 2w + 2 * w 96 ≥ 4w + 2w 96 ≥ 6w 96 ÷ 6 ≥ 6w ÷ 6
(Example 8 - continued) Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. Maximum Perimeter ≥ 2l + 2w 96 ≥ 2 * 2w + 2 * w 96 ≥ 4w + 2w 96 ≥ 6w 96 ÷ 6 ≥ 6w ÷ 6 16 ≥ w
(Example 8 - continued) Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. Maximum Perimeter ≥ 2l + 2w 96 ≥ 2 * 2w + 2 * w 96 ≥ 4w + 2w 96 ≥ 6w 96 ÷ 6 ≥ 6w ÷ 6 16 ≥ w Jim can fence a space with a width less than or equal to 16 feet.

More Related Content

KEY
5-11 Relating Multiplication and Division
Rudy Alfonso
 
PDF
Day 12 variables on both sides fractions
Erik Tjersland
 
PDF
Day 12 variables on both sides fractions
Erik Tjersland
 
KEY
Unit 3 hw 2 - solving 1 step equations
Lori Rapp
 
PDF
Day 5 fraction two step
Erik Tjersland
 
PPTX
3.5 & 3.6 writing and solving ratios and proportions day 2
bweldon
 
KEY
Unit 3 hw 4 - solving equations variable both sides
Lori Rapp
 
PDF
Day 2 mult div
Erik Tjersland
 
5-11 Relating Multiplication and Division
Rudy Alfonso
 
Day 12 variables on both sides fractions
Erik Tjersland
 
Day 12 variables on both sides fractions
Erik Tjersland
 
Unit 3 hw 2 - solving 1 step equations
Lori Rapp
 
Day 5 fraction two step
Erik Tjersland
 
3.5 & 3.6 writing and solving ratios and proportions day 2
bweldon
 
Unit 3 hw 4 - solving equations variable both sides
Lori Rapp
 
Day 2 mult div
Erik Tjersland
 

Viewers also liked (9)

DOC
SHOAIB
Muhammad Shoaib
 
PDF
AEED
Oleg Nikulin, PhD in Engineering Science
 
PPT
Интеграция удаленного доступа к регулируемым электроприводам буровых механизм...
Oleg Nikulin, PhD in Engineering Science
 
PDF
iWizard_Report_MarynaRazakhatskaya
Marina Razakhatskaya
 
PDF
Millen Capital Brochure
Angus Millen
 
DOCX
Abstrak
Sriwijaya University
 
PDF
garcia_reporter_resume
Isabel Garcia
 
PPTX
Situación didáctica
Diana Reyes
 
PPTX
El poder de los medios
Isis Andrea Aku
 
SHOAIB
Muhammad Shoaib
 
AEED
Oleg Nikulin, PhD in Engineering Science
 
Интеграция удаленного доступа к регулируемым электроприводам буровых механизм...
Oleg Nikulin, PhD in Engineering Science
 
iWizard_Report_MarynaRazakhatskaya
Marina Razakhatskaya
 
Millen Capital Brochure
Angus Millen
 
Abstrak
Sriwijaya University
 
garcia_reporter_resume
Isabel Garcia
 
Situación didáctica
Diana Reyes
 
El poder de los medios
Isis Andrea Aku
 
Ad

Similar to Solving Inequalities Notes (20)

PPTX
Solving One-Step Equations
vanessadalton
 
PDF
Topic1 equations student
mramithnair
 
PDF
Day 5 combining like terms equations
Erik Tjersland
 
PPT
Solving equations variable both sides
carolinevest77
 
PPT
31 32review Equation Steps
taco40
 
PPT
Multi-Step equations with fractions and decimals _1_ (1).ppt
JoseTorres185927
 
PPT
1.7 solving absolute value inequalities
fthrower
 
PPTX
Wk5wednesday
caronbyrd
 
PPT
Inequalities review
Olivia Dombrowski
 
PDF
Day 9 combining like terms
Erik Tjersland
 
KEY
Compound Inequalities Notes
Lori Rapp
 
PDF
Day 5 combining like terms equations
Erik Tjersland
 
PDF
D ay 1 add subt
Erik Tjersland
 
KEY
5-11 Relating Multiplication and Division
Mel Anthony Pepito
 
PDF
Day 10 variables on both sides
Erik Tjersland
 
PDF
Day 10 variables on both sides
Erik Tjersland
 
PPTX
3 2, 3-3 solving one-step inequalities
hisema01
 
PDF
D ay 1 add subt
Erik Tjersland
 
PDF
Day 10 variables on both sides
Erik Tjersland
 
PPTX
Solving all equations with sound
robertleichner
 
Solving One-Step Equations
vanessadalton
 
Topic1 equations student
mramithnair
 
Day 5 combining like terms equations
Erik Tjersland
 
Solving equations variable both sides
carolinevest77
 
31 32review Equation Steps
taco40
 
Multi-Step equations with fractions and decimals _1_ (1).ppt
JoseTorres185927
 
1.7 solving absolute value inequalities
fthrower
 
Wk5wednesday
caronbyrd
 
Inequalities review
Olivia Dombrowski
 
Day 9 combining like terms
Erik Tjersland
 
Compound Inequalities Notes
Lori Rapp
 
Day 5 combining like terms equations
Erik Tjersland
 
D ay 1 add subt
Erik Tjersland
 
5-11 Relating Multiplication and Division
Mel Anthony Pepito
 
Day 10 variables on both sides
Erik Tjersland
 
Day 10 variables on both sides
Erik Tjersland
 
3 2, 3-3 solving one-step inequalities
hisema01
 
D ay 1 add subt
Erik Tjersland
 
Day 10 variables on both sides
Erik Tjersland
 
Solving all equations with sound
robertleichner
 
Ad

More from Lori Rapp (20)

PDF
Piecewise functions
Lori Rapp
 
PDF
Normal curve
Lori Rapp
 
PDF
Venn diagrams
Lori Rapp
 
PPT
Circles notes
Lori Rapp
 
PPT
Quadrilateral notes
Lori Rapp
 
KEY
Remainder & Factor Theorems
Lori Rapp
 
KEY
Multiplying polynomials - part 1
Lori Rapp
 
KEY
Develop the Area of a Circle Formula
Lori Rapp
 
KEY
Unit 4 hw 8 - pointslope, parallel & perp
Lori Rapp
 
KEY
Sets Notes
Lori Rapp
 
KEY
Absolute Value Inequalities Notes
Lori Rapp
 
KEY
Solving quadratic equations part 1
Lori Rapp
 
KEY
Introduction to Equations Notes
Lori Rapp
 
KEY
Associative property
Lori Rapp
 
PDF
Real numbers
Lori Rapp
 
KEY
Unit 4 hw 7 - direct variation & linear equation give 2 points
Lori Rapp
 
KEY
Absolute Value Equations
Lori Rapp
 
KEY
Unit 3 hw 7 - literal equations
Lori Rapp
 
KEY
Unit 4 hw 1 - coordinate plane, d&r, midpoint
Lori Rapp
 
KEY
Unit 2 hw5 mult fraction, simplify
Lori Rapp
 
Piecewise functions
Lori Rapp
 
Normal curve
Lori Rapp
 
Venn diagrams
Lori Rapp
 
Circles notes
Lori Rapp
 
Quadrilateral notes
Lori Rapp
 
Remainder & Factor Theorems
Lori Rapp
 
Multiplying polynomials - part 1
Lori Rapp
 
Develop the Area of a Circle Formula
Lori Rapp
 
Unit 4 hw 8 - pointslope, parallel & perp
Lori Rapp
 
Sets Notes
Lori Rapp
 
Absolute Value Inequalities Notes
Lori Rapp
 
Solving quadratic equations part 1
Lori Rapp
 
Introduction to Equations Notes
Lori Rapp
 
Associative property
Lori Rapp
 
Real numbers
Lori Rapp
 
Unit 4 hw 7 - direct variation & linear equation give 2 points
Lori Rapp
 
Absolute Value Equations
Lori Rapp
 
Unit 3 hw 7 - literal equations
Lori Rapp
 
Unit 4 hw 1 - coordinate plane, d&r, midpoint
Lori Rapp
 
Unit 2 hw5 mult fraction, simplify
Lori Rapp
 

Recently uploaded (20)

PDF
FourierSeries-QuestionsWithAnswers(Part-A).pdf
Raji Murugan
 
PDF
Supply Chain Operations Speaking Notes -ICLT Program
labyankof
 
PPTX
PPH.pptx obstetrics and gynecology in nursing
SrideviDevaraj5
 
PDF
2.FourierTransform-ShortQuestionswithAnswers.pdf
Raji Murugan
 
PDF
O5-L3 Freight Transport Ops (International) V1.pdf
labyankof
 
PPTX
Introduction_to_Human_Anatomy_and_Physiology_for_B.Pharm.pptx
chetansingh379583
 
PPTX
Renaissance Architecture: A Journey from Faith to Humanism
DrMonicaSharma
 
PPTX
PPT- ENG7_QUARTER1_LESSON1_WEEK1. IMAGERY -DESCRIPTIONS pptx.pptx
KMDee
 
PDF
Sports Quiz easy sports quiz sports quiz
nikunj2757
 
PDF
Abdominal Access Techniques with Prof. Dr. R K Mishra
mohitsuren827
 
PDF
Computing-Curriculum for Schools in Ghana
abigailanane203
 
PDF
Saundersa Comprehensive Review for the NCLEX-RN Examination.pdf
sreejithcareers
 
PDF
O7-L3 Supply Chain Operations - ICLT Program
labyankof
 
PDF
VCE English Exam - Section C Student Revision Booklet
jpinnuck
 
PDF
102 student loan defaulters named and shamed – Is someone you know on the list?
Kweku Zurek
 
PDF
ANTIBIOTICS.pptx.pdf………………… xxxxxxxxxxxxx
HakHe
 
PDF
Insiders guide to clinical Medicine.pdf
AbhishekPatil315128
 
PDF
Complications of Minimal Access Surgery at WLH
mohitsuren827
 
PPTX
Pharma ospi slides which help in ospi learning
rameetkumar304
 
PDF
TR - Agricultural Crops Production NC III.pdf
avgilmegino3
 
FourierSeries-QuestionsWithAnswers(Part-A).pdf
Raji Murugan
 
Supply Chain Operations Speaking Notes -ICLT Program
labyankof
 
PPH.pptx obstetrics and gynecology in nursing
SrideviDevaraj5
 
2.FourierTransform-ShortQuestionswithAnswers.pdf
Raji Murugan
 
O5-L3 Freight Transport Ops (International) V1.pdf
labyankof
 
Introduction_to_Human_Anatomy_and_Physiology_for_B.Pharm.pptx
chetansingh379583
 
Renaissance Architecture: A Journey from Faith to Humanism
DrMonicaSharma
 
PPT- ENG7_QUARTER1_LESSON1_WEEK1. IMAGERY -DESCRIPTIONS pptx.pptx
KMDee
 
Sports Quiz easy sports quiz sports quiz
nikunj2757
 
Abdominal Access Techniques with Prof. Dr. R K Mishra
mohitsuren827
 
Computing-Curriculum for Schools in Ghana
abigailanane203
 
Saundersa Comprehensive Review for the NCLEX-RN Examination.pdf
sreejithcareers
 
O7-L3 Supply Chain Operations - ICLT Program
labyankof
 
VCE English Exam - Section C Student Revision Booklet
jpinnuck
 
102 student loan defaulters named and shamed – Is someone you know on the list?
Kweku Zurek
 
ANTIBIOTICS.pptx.pdf………………… xxxxxxxxxxxxx
HakHe
 
Insiders guide to clinical Medicine.pdf
AbhishekPatil315128
 
Complications of Minimal Access Surgery at WLH
mohitsuren827
 
Pharma ospi slides which help in ospi learning
rameetkumar304
 
TR - Agricultural Crops Production NC III.pdf
avgilmegino3
 

Solving Inequalities Notes

  • 2. The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks?
  • 3. The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Think about how to solve this problem.
  • 4. The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Think about how to solve this problem. • What number did you come up with?
  • 5. The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Think about how to solve this problem. • What number did you come up with? • Ask yourself if the number makes sense in the context of this problem.
  • 6. The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Think about how to solve this problem. • What number did you come up with? • Ask yourself if the number makes sense in the context of this problem. • How did you solve this problem?
  • 7. The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Think about how to solve this problem. • What number did you come up with? • Ask yourself if the number makes sense in the context of this problem. • How did you solve this problem? • Keep your answers and we’ll return to this problem later.
  • 8. Recall Solving Equations 2x − 7 = 11
  • 9. Recall Solving Equations • Use the properties of equality to 2x − 7 = 11 isolate the variable.
  • 10. Recall Solving Equations • Use the properties of equality to 2x − 7 = 11 isolate the variable. • Undo subtraction (or addition) first using the opposite operation.
  • 11. Recall Solving Equations • Use the properties of equality to 2x − 7 = 11 isolate the variable. +7 +7 • Undo subtraction (or addition) first using the opposite operation.
  • 12. Recall Solving Equations • Use the properties of equality to 2x − 7 = 11 isolate the variable. +7 +7 • Undo subtraction (or addition) first using the opposite operation. 2x = 18
  • 13. Recall Solving Equations • Use the properties of equality to 2x − 7 = 11 isolate the variable. +7 +7 • Undo subtraction (or addition) first using the opposite operation. 2x = 18 • Then undo multiplication (or division) using the opposite operation.
  • 14. Recall Solving Equations • Use the properties of equality to 2x − 7 = 11 isolate the variable. +7 +7 • Undo subtraction (or addition) first using the opposite operation. 2x = 18 • Then undo multiplication (or division) 2 2 using the opposite operation.
  • 15. Recall Solving Equations • Use the properties of equality to 2x − 7 = 11 isolate the variable. +7 +7 • Undo subtraction (or addition) first using the opposite operation. 2x = 18 • Then undo multiplication (or division) 2 2 using the opposite operation. x=9
  • 16. Recall Solving Equations • Use the properties of equality to 2x − 7 = 11 isolate the variable. +7 +7 • Undo subtraction (or addition) first using the opposite operation. 2x = 18 • Then undo multiplication (or division) 2 2 using the opposite operation. x=9 • The coefficient of x is 1. Check your solution in the original problem.
  • 17. Recall Solving Equations • Use the properties of equality to 2x − 7 = 11 isolate the variable. +7 +7 • Undo subtraction (or addition) first using the opposite operation. 2x = 18 • Then undo multiplication (or division) 2 2 using the opposite operation. x=9 • The coefficient of x is 1. Check your solution in the original problem. Check: ? 2 ⋅ 9 − 7 = 11 11 = 11
  • 18. Recall Solving Equations • Use the properties of equality to 2x − 7 = 11 isolate the variable. +7 +7 • Undo subtraction (or addition) first using the opposite operation. 2x = 18 • Then undo multiplication (or division) 2 2 using the opposite operation. x=9 • The coefficient of x is 1. Check your solution in the original problem. Check: ? • It works so your solution is correct. 2 ⋅ 9 − 7 = 11 11 = 11
  • 19. Solving Inequality - Example 1 2x − 7 > 11
  • 20. Solving Inequality - Example 1 • Use the properties of equality to isolate the variable. 2x − 7 > 11
  • 21. Solving Inequality - Example 1 • Use the properties of equality to isolate the variable. 2x − 7 > 11 • Undo subtraction (or addition) first using the opposite operation.
  • 22. Solving Inequality - Example 1 • Use the properties of equality to isolate the variable. 2x − 7 > 11 • Undo subtraction (or addition) first using the +7 +7 opposite operation.
  • 23. Solving Inequality - Example 1 • Use the properties of equality to isolate the variable. 2x − 7 > 11 • Undo subtraction (or addition) first using the +7 +7 opposite operation. 2x > 18
  • 24. Solving Inequality - Example 1 • Use the properties of equality to isolate the variable. 2x − 7 > 11 • Undo subtraction (or addition) first using the +7 +7 opposite operation. 2x > 18 • Then undo multiplication (or division) using the opposite operation.
  • 25. Solving Inequality - Example 1 • Use the properties of equality to isolate the variable. 2x − 7 > 11 • Undo subtraction (or addition) first using the +7 +7 opposite operation. 2x > 18 • Then undo multiplication (or division) using 2 2 the opposite operation.
  • 26. Solving Inequality - Example 1 • Use the properties of equality to isolate the variable. 2x − 7 > 11 • Undo subtraction (or addition) first using the +7 +7 opposite operation. 2x > 18 • Then undo multiplication (or division) using 2 2 the opposite operation. x>9
  • 27. Solving Inequality - Example 1 • Use the properties of equality to isolate the variable. 2x − 7 > 11 • Undo subtraction (or addition) first using the +7 +7 opposite operation. 2x > 18 • Then undo multiplication (or division) using 2 2 the opposite operation. • The coefficient of x is 1. x>9
  • 28. Solving Inequality - Example 1 • Use the properties of equality to isolate the variable. 2x − 7 > 11 • Undo subtraction (or addition) first using the +7 +7 opposite operation. 2x > 18 • Then undo multiplication (or division) using 2 2 the opposite operation. • The coefficient of x is 1. x>9 • Steps were the same!
  • 29. Solving Inequality - Example 1 • Use the properties of equality to isolate the variable. 2x − 7 > 11 • Undo subtraction (or addition) first using the +7 +7 opposite operation. 2x > 18 • Then undo multiplication (or division) using 2 2 the opposite operation. • The coefficient of x is 1. x>9 • Steps were the same! • Final step is to check the solution.
  • 30. Example 1 (con’t) • Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10
  • 31. Example 1 (con’t) • Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10 • Open circle at 9 because >, not ≥.
  • 32. Example 1 (con’t) • Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10 • Open circle at 9 because >, not ≥. • Always shade in the direction of a number that works in the inequality.
  • 33. Example 1 (con’t) • Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10 • Open circle at 9 because >, not ≥. • Always shade in the direction of a number that works in the inequality. • Shade to the right because 10 > 9.
  • 34. Example 1 (con’t) • Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10 • Open circle at 9 because >, not ≥. • Always shade in the direction of a number that works in the inequality. • Shade to the right because 10 > 9. • Check the solution in the ORIGINAL problem.
  • 35. Example 1 (con’t) • Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10 • Open circle at 9 because >, not ≥. • Always shade in the direction of a number that works in the inequality. Original Problem: • Shade to the right because 10 > 9. 2x − 7 > 11 • Check the solution in the ORIGINAL problem.
  • 36. Example 1 (con’t) • Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10 • Open circle at 9 because >, not ≥. • Always shade in the direction of a number that works in the inequality. Original Problem: • Shade to the right because 10 > 9. 2x − 7 > 11 • Check the solution in the ORIGINAL problem. • Check the solution 10.
  • 37. Example 1 (con’t) • Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10 • Open circle at 9 because >, not ≥. • Always shade in the direction of a number that works in the inequality. Original Problem: • Shade to the right because 10 > 9. 2x − 7 > 11 • Check the solution in the ORIGINAL Check: problem. ? 2 ⋅10 − 7 > 11 • Check the solution 10. 13 > 11
  • 38. Example 1 (con’t) • Graph the solution to help with the x>9 check. 0 1 2 3 4 5 6 7 8 9 10 • Open circle at 9 because >, not ≥. • Always shade in the direction of a number that works in the inequality. Original Problem: • Shade to the right because 10 > 9. 2x − 7 > 11 • Check the solution in the ORIGINAL Check: problem. ? 2 ⋅10 − 7 > 11 • Check the solution 10. 13 > 11 • This is true and therefore a solution.
  • 39. Example 1 (con’t) • It is good to check at least 2 other numbers x>9 for inequalities. 0 1 2 3 4 5 6 7 8 9 10 Original Problem: 2x − 7 > 11 Check:
  • 40. Example 1 (con’t) • It is good to check at least 2 other numbers x>9 for inequalities. 0 1 2 3 4 5 6 7 8 9 10 • Pick a number that is not a solution and the number where the circle is placed. Original Problem: 2x − 7 > 11 Check:
  • 41. Example 1 (con’t) • It is good to check at least 2 other numbers x>9 for inequalities. 0 1 2 3 4 5 6 7 8 9 10 • Pick a number that is not a solution and the number where the circle is placed. Original Problem: • 0 is not a solution. 2x − 7 > 11 Check:
  • 42. Example 1 (con’t) • It is good to check at least 2 other numbers x>9 for inequalities. 0 1 2 3 4 5 6 7 8 9 10 • Pick a number that is not a solution and the number where the circle is placed. Original Problem: • 0 is not a solution. 2x − 7 > 11 • The check shows -7 > 11, which is not true. This is good because it is not shaded. Check: ? 2 ⋅ 0 − 7 > 11 −7 > 11 x
  • 43. Example 1 (con’t) • It is good to check at least 2 other numbers x>9 for inequalities. 0 1 2 3 4 5 6 7 8 9 10 • Pick a number that is not a solution and the number where the circle is placed. Original Problem: • 0 is not a solution. 2x − 7 > 11 • The check shows -7 > 11, which is not true. This is good because it is not shaded. Check: ? • 9 is not a solution because of the open 2 ⋅ 0 − 7 > 11 circle. −7 > 11 x
  • 44. Example 1 (con’t) • It is good to check at least 2 other numbers x>9 for inequalities. 0 1 2 3 4 5 6 7 8 9 10 • Pick a number that is not a solution and the number where the circle is placed. Original Problem: • 0 is not a solution. 2x − 7 > 11 • The check shows -7 > 11, which is not true. This is good because it is not shaded. Check: ? • 9 is not a solution because of the open 2 ⋅ 0 − 7 > 11 circle. −7 > 11 x ? • 11 is not greater than 11. Again this is 2 ⋅ 9 − 7 > 11 good because 9 is not a solution. 11 > 11 x
  • 46. Solving Inequality Recap 1) Solve the inequality as you would an equation.
  • 47. Solving Inequality Recap 1) Solve the inequality as you would an equation. 2) Graph the solution on a number line.
  • 48. Solving Inequality Recap 1) Solve the inequality as you would an equation. 2) Graph the solution on a number line. 3) Check at least 3 numbers in the original inequality.
  • 49. Solving Inequality Recap 1) Solve the inequality as you would an equation. 2) Graph the solution on a number line. 3) Check at least 3 numbers in the original inequality. 4) Write the solution.
  • 50. Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. • Check solution in original inequality. • -5 is NOT greater than -3! • Why didn’t it work? • Negative means reverse. When dividing by a negative, don’t we reverse all the signs? • The same goes for the inequality. When dividing (or multiplying) by a negative, the inequality must reverse.
  • 51. Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. −1 −1 • Check solution in original inequality. • -5 is NOT greater than -3! • Why didn’t it work? • Negative means reverse. When dividing by a negative, don’t we reverse all the signs? • The same goes for the inequality. When dividing (or multiplying) by a negative, the inequality must reverse.
  • 52. Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. −1 −1 • Check solution in original inequality. • -5 is NOT greater than -3! −2x > −4 • Why didn’t it work? • Negative means reverse. When dividing by a negative, don’t we reverse all the signs? • The same goes for the inequality. When dividing (or multiplying) by a negative, the inequality must reverse.
  • 53. Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. −1 −1 • Check solution in original inequality. • -5 is NOT greater than -3! −2x > −4 • Why didn’t it work? -2 -2 • Negative means reverse. When dividing by a negative, don’t we reverse all the signs? • The same goes for the inequality. When dividing (or multiplying) by a negative, the inequality must reverse.
  • 54. Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. −1 −1 • Check solution in original inequality. • -5 is NOT greater than -3! −2x > −4 • Why didn’t it work? -2 -2 • Negative means reverse. When dividing x>2 by a negative, don’t we reverse all the signs? • The same goes for the inequality. When dividing (or multiplying) by a negative, the inequality must reverse.
  • 55. Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. −1 −1 −2x > −4 -2 -2 x>2 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 56. Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. −1 −1 −2x > −4 -2 -2 x>2 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 57. Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. −1 −1 • Check solution in original inequality. −2x > −4 -2 -2 x>2 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 58. Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. −1 −1 • Check solution in original inequality. −2x > −4 -2 -2 x>2 -5 -4 -3 -2 -1 0 1 2 3 4 5 Check: ? −2 ⋅ 3 + 1>− 3 −5 > −3
  • 59. Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. −1 −1 • Check solution in original inequality. • -5 is NOT greater than -3! −2x > −4 -2 -2 x>2 -5 -4 -3 -2 -1 0 1 2 3 4 5 Check: ? −2 ⋅ 3 + 1>− 3 −5 > −3 x
  • 60. Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. −1 −1 • Check solution in original inequality. • -5 is NOT greater than -3! −2x > −4 • Why didn’t it work? -2 -2 x>2 -5 -4 -3 -2 -1 0 1 2 3 4 5 Check: ? −2 ⋅ 3 + 1>− 3 −5 > −3 x
  • 61. Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. −1 −1 • Check solution in original inequality. • -5 is NOT greater than -3! −2x > −4 • Why didn’t it work? -2 -2 • Negative means reverse. When dividing x>2 by a negative, don’t we reverse all the -5 -4 -3 -2 -1 0 1 2 3 4 5 signs? Check: ? −2 ⋅ 3 + 1>− 3 −5 > −3 x
  • 62. Example 2 - What’s wrong here... • Solve for x. −2x + 1 > −3 • Graph. −1 −1 • Check solution in original inequality. • -5 is NOT greater than -3! −2x > −4 • Why didn’t it work? -2 -2 • Negative means reverse. When dividing x>2 by a negative, don’t we reverse all the -5 -4 -3 -2 -1 0 1 2 3 4 5 signs? Check: ? • The same goes for the inequality. When −2 ⋅ 3 + 1>− 3 dividing (or multiplying) by a negative, the inequality must reverse. −5 > −3 x
  • 63. Example 2 (con’t) • Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. • Graph. • Check solution in the original inequality. • 1 is greater than -3! • Write the solution.
  • 64. Example 2 (con’t) • Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1 • Graph. • Check solution in the original inequality. • 1 is greater than -3! • Write the solution.
  • 65. Example 2 (con’t) • Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1 • Graph. −2x > −4 • Check solution in the original inequality. • 1 is greater than -3! • Write the solution.
  • 66. Example 2 (con’t) • Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1 • Graph. −2x > −4 -2 -2 • Check solution in the original inequality. • 1 is greater than -3! • Write the solution.
  • 67. Example 2 (con’t) • Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1 • Graph. −2x > −4 -2 -2 • Check solution in the original x<2 inequality. • 1 is greater than -3! • Write the solution.
  • 68. Example 2 (con’t) • Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1 • Graph. −2x > −4 -2 -2 x<2
  • 69. Example 2 (con’t) • Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1 • Graph. −2x > −4 -2 -2 x<2 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 70. Example 2 (con’t) • Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1 • Graph. −2x > −4 -2 -2 • Check solution in the original x<2 inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 71. Example 2 (con’t) • Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1 • Graph. −2x > −4 -2 -2 • Check solution in the original x<2 inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 Check: ? −2 ⋅ 0 + 1>− 3 1 > −3
  • 72. Example 2 (con’t) • Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1 • Graph. −2x > −4 -2 -2 • Check solution in the original x<2 inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 • 1 is greater than -3! Check: ? −2 ⋅ 0 + 1>− 3 1 > −3
  • 73. Example 2 (con’t) • Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1 • Graph. −2x > −4 -2 -2 • Check solution in the original x<2 inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 • 1 is greater than -3! Check: ? • Write the solution. −2 ⋅ 0 + 1>− 3 1 > −3
  • 74. Example 2 (con’t) • Solve for x but reverse the −2x + 1 > −3 inequality when you divide by -2. −1 −1 • Graph. −2x > −4 -2 -2 • Check solution in the original x<2 inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 • 1 is greater than -3! Check: ? • Write the solution. −2 ⋅ 0 + 1>− 3 {x | x < 2} 1 > −3
  • 76. Equation vs Inequality • What are the similarities between solving an Equation and solving an Inequality?
  • 77. Equation vs Inequality • What are the similarities between solving an Equation and solving an Inequality? ✓ Solve using the same process.
  • 78. Equation vs Inequality • What are the similarities between solving an Equation and solving an Inequality? ✓ Solve using the same process. ✓ Always check solution.
  • 80. Equation vs Inequality • What are the differences between solving an Equation and solving an Inequality?
  • 81. Equation vs Inequality • What are the differences between solving an Equation and solving an Inequality? ✓ When multiplying or dividing by a negative, the inequality reverses.
  • 82. Equation vs Inequality • What are the differences between solving an Equation and solving an Inequality? ✓ When multiplying or dividing by a negative, the inequality reverses. ‣ < changes to >
  • 83. Equation vs Inequality • What are the differences between solving an Equation and solving an Inequality? ✓ When multiplying or dividing by a negative, the inequality reverses. ‣ < changes to > ‣ > changes to <
  • 84. Equation vs Inequality • What are the differences between solving an Equation and solving an Inequality? ✓ When multiplying or dividing by a negative, the inequality reverses. ‣ < changes to > ‣ > changes to < ‣ ≤ changes to ≥
  • 85. Equation vs Inequality • What are the differences between solving an Equation and solving an Inequality? ✓ When multiplying or dividing by a negative, the inequality reverses. ‣ < changes to > ‣ > changes to < ‣ ≤ changes to ≥ ‣ ≥ changes to ≤
  • 86. Equation vs Inequality • What are the differences between solving an Equation and solving an Inequality? ✓ When multiplying or dividing by a negative, the inequality reverses. ‣ < changes to > ‣ > changes to < ‣ ≤ changes to ≥ ‣ ≥ changes to ≤ ✓ Check multiple numbers to include solutions and non- solutions.
  • 87. Example 3 - You try... • Solve, graph, check, and write your solution to 8x − 11 ≤ 13
  • 88. Example 3 - You try... • Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11
  • 89. Example 3 - You try... • Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 8x ≤ 24
  • 90. Example 3 - You try... • Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 8x ≤ 24 8 8
  • 91. Example 3 - You try... • Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 8x ≤ 24 8 8 x≤3
  • 92. Example 3 - You try... • Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 -5 -4 -3 -2 -1 0 1 2 3 4 5 8x ≤ 24 8 8 x≤3
  • 93. Example 3 - You try... • Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 -5 -4 -3 -2 -1 0 1 2 3 4 5 8x ≤ 24 8 8 x≤3
  • 94. Example 3 - You try... • Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 -5 -4 -3 -2 -1 0 1 2 3 4 5 8x ≤ 24 Check: 8 8 x≤3
  • 95. Example 3 - You try... • Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 -5 -4 -3 -2 -1 0 1 2 3 4 5 8x ≤ 24 Check: ? 8 8 8 ⋅ 0 − 11≤ 13 −11 ≤ 13 x≤3
  • 96. Example 3 - You try... • Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 -5 -4 -3 -2 -1 0 1 2 3 4 5 8x ≤ 24 Check: ? ? 8 8 8 ⋅ 0 − 11≤ 13 8 ⋅ 5 − 11≤ 13 x≤3 −11 ≤ 13 29 ≤ 13 x
  • 97. Example 3 - You try... • Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 -5 -4 -3 -2 -1 0 1 2 3 4 5 8x ≤ 24 Check: ? ? 8 8 8 ⋅ 0 − 11≤ 13 8 ⋅ 5 − 11≤ 13 x≤3 −11 ≤ 13 ? 29 ≤ 13 x 8 ⋅ 3 − 11≤ 13 13 ≤ 13
  • 98. Example 3 - You try... • Solve, graph, check, and write your solution to 8x − 11 ≤ 13 +11 +11 -5 -4 -3 -2 -1 0 1 2 3 4 5 8x ≤ 24 Check: ? ? 8 8 8 ⋅ 0 − 11≤ 13 8 ⋅ 5 − 11≤ 13 x≤3 −11 ≤ 13 ? 29 ≤ 13 x 8 ⋅ 3 − 11≤ 13 13 ≤ 13 {x | x ≤ 3}
  • 99. Example 4 - You try... • Solve, graph, check, and write your solution to −3x + 10 > −5
  • 100. Example 4 - You try... • Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10
  • 101. Example 4 - You try... • Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 −3x > −15
  • 102. Example 4 - You try... • Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 −3x > −15 −3 −3
  • 103. Example 4 - You try... • Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 −3x > −15 −3 −3 x<5
  • 104. Example 4 - You try... • Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 -5 -4 -3 -2 -1 0 1 2 3 4 5 −3x > −15 −3 −3 x<5
  • 105. Example 4 - You try... • Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 -5 -4 -3 -2 -1 0 1 2 3 4 5 −3x > −15 −3 −3 x<5
  • 106. Example 4 - You try... • Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 -5 -4 -3 -2 -1 0 1 2 3 4 5 −3x > −15 Check: −3 −3 x<5
  • 107. Example 4 - You try... • Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 -5 -4 -3 -2 -1 0 1 2 3 4 5 −3x > −15 Check: ? −3 −3 −3 ⋅ 0 + 10 >− 5 10 > −5 x<5
  • 108. Example 4 - You try... • Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 -5 -4 -3 -2 -1 0 1 2 3 4 5 −3x > −15 Check: ? ? −3 −3 −3 ⋅ 0 + 10 >− 5 −3 ⋅10 + 10 >− 5 x<5 10 > −5 −20 > −5 x
  • 109. Example 4 - You try... • Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 -5 -4 -3 -2 -1 0 1 2 3 4 5 −3x > −15 Check: ? ? −3 −3 −3 ⋅ 0 + 10 >− 5 −3 ⋅10 + 10 >− 5 x<5 10 > −5 −20 > −5 x ? −3 ⋅ 5 + 10 >− 5 −5 > −5 x
  • 110. Example 4 - You try... • Solve, graph, check, and write your solution to −3x + 10 > −5 −10 −10 -5 -4 -3 -2 -1 0 1 2 3 4 5 −3x > −15 Check: ? ? −3 −3 −3 ⋅ 0 + 10 >− 5 −3 ⋅10 + 10 >− 5 x<5 10 > −5 −20 > −5 x ? −3 ⋅ 5 + 10 >− 5 {x | x < 5} −5 > −5 x
  • 111. Graphing Caution! • Did you notice that the shading always went in the direction of the inequality?
  • 112. Graphing Caution! • Did you notice that the shading always went in the direction of the inequality? • Graph 1 > x
  • 113. Graphing Caution! • Did you notice that the shading always went in the direction of the inequality? • Graph 1 > x -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 114. Graphing Caution! • Did you notice that the shading always went in the direction of the inequality? • Graph 1 > x -5 -4 -3 -2 -1 0 1 2 3 4 5 • Notice the shading is opposite the direction of the inequality. This is because x is on the right of the symbol instead of the left.
  • 115. Graphing Caution! • Did you notice that the shading always went in the direction of the inequality? • Graph 1 > x -5 -4 -3 -2 -1 0 1 2 3 4 5 • Notice the shading is opposite the direction of the inequality. This is because x is on the right of the symbol instead of the left. • Always check numbers when determining shading!
  • 116. What if parenthesis and/or variables on both sides of the inequality?
  • 117. What if parenthesis and/or variables on both sides of the inequality? • Solve the same as you would if it was an equation.
  • 118. What if parenthesis and/or variables on both sides of the inequality? • Solve the same as you would if it was an equation. • First, remove parenthesis.
  • 119. What if parenthesis and/or variables on both sides of the inequality? • Solve the same as you would if it was an equation. • First, remove parenthesis. • Second, combine like terms on the same side of the inequality (if any).
  • 120. What if parenthesis and/or variables on both sides of the inequality? • Solve the same as you would if it was an equation. • First, remove parenthesis. • Second, combine like terms on the same side of the inequality (if any). • Third, get variables on the same side of inequality.
  • 121. What if parenthesis and/or variables on both sides of the inequality? • Solve the same as you would if it was an equation. • First, remove parenthesis. • Second, combine like terms on the same side of the inequality (if any). • Third, get variables on the same side of inequality. • Finally, solve for the variable.
  • 122. Example 4 - 1− ( 4 + 3x ) < −3 − 5x
  • 123. Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x
  • 124. Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x 1− 4 − 3x < −3 − 5x
  • 125. Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x • Combine like terms on the 1− 4 − 3x < −3 − 5x left.
  • 126. Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x • Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x
  • 127. Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x • Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x • Move variables on one side.
  • 128. Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x • Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x • Move variables on one side. +5x +5x
  • 129. Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x • Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x • Move variables on one side. +5x +5x • Solve for x.
  • 130. Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x • Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x • Move variables on one side. +5x +5x • Solve for x. −3 + 2x < −3
  • 131. Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x • Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x • Move variables on one side. +5x +5x • Solve for x. −3 + 2x < −3 +3 +3
  • 132. Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x • Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x • Move variables on one side. +5x +5x • Solve for x. −3 + 2x < −3 +3 +3 2x < 0
  • 133. Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x • Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x • Move variables on one side. +5x +5x • Solve for x. −3 + 2x < −3 +3 +3 2x < 0 2 2
  • 134. Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x • Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x • Move variables on one side. +5x +5x • Solve for x. −3 + 2x < −3 +3 +3 2x < 0 2 2 x<0
  • 135. Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x • Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x • Move variables on one side. +5x +5x • Solve for x. −3 + 2x < −3 +3 +3 • Graph solution. 2x < 0 2 2 x<0
  • 136. Example 4 - • Remove parenthesis. 1− ( 4 + 3x ) < −3 − 5x • Combine like terms on the 1− 4 − 3x < −3 − 5x left. −3 − 3x < −3 − 5x • Move variables on one side. +5x +5x • Solve for x. −3 + 2x < −3 +3 +3 • Graph solution. 2x < 0 2 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 x<0
  • 137. Example 4 Continued... • Check at least 3 solutions 1− ( 4 + 3x ) < −3 − 5x in original inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 • The solution checks out.
  • 138. Example 4 Continued... • Check at least 3 solutions 1− ( 4 + 3x ) < −3 − 5x in original inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 Check: • The solution checks out.
  • 139. Example 4 Continued... • Check at least 3 solutions 1− ( 4 + 3x ) < −3 − 5x in original inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 ? Check: 1− ( 4 + 3⋅ −1) <− 3 − 5 ⋅ −1 1− 1 < −3 + 5 0<2 • The solution checks out.
  • 140. Example 4 Continued... • Check at least 3 solutions 1− ( 4 + 3x ) < −3 − 5x in original inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 ? Check: ? 1− ( 4 + 3⋅ −1) <− 3 − 5 ⋅ −1 1− ( 4 + 3⋅1) <− 3 − 5 ⋅1 1− 1 < −3 + 5 1− 7 < −3 − 5 0<2 −6 < −8 x • The solution checks out.
  • 141. Example 4 Continued... • Check at least 3 solutions 1− ( 4 + 3x ) < −3 − 5x in original inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 ? Check: ? 1− ( 4 + 3⋅ −1) <− 3 − 5 ⋅ −1 1− ( 4 + 3⋅1) <− 3 − 5 ⋅1 1− 1 < −3 + 5 1− 7 < −3 − 5 0<2 ? −6 < −8 x 1− ( 4 + 3⋅ 0 ) <− 3 − 5 ⋅ 0 −3 < −3 x • The solution checks out.
  • 142. Example 4 Continued... • Check at least 3 solutions 1− ( 4 + 3x ) < −3 − 5x in original inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 ? Check: ? 1− ( 4 + 3⋅ −1) <− 3 − 5 ⋅ −1 1− ( 4 + 3⋅1) <− 3 − 5 ⋅1 1− 1 < −3 + 5 1− 7 < −3 − 5 0<2 ? −6 < −8 x 1− ( 4 + 3⋅ 0 ) <− 3 − 5 ⋅ 0 −3 < −3 x • The solution checks out.
  • 143. Example 4 Continued... • Check at least 3 solutions 1− ( 4 + 3x ) < −3 − 5x in original inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 ? Check: ? 1− ( 4 + 3⋅ −1) <− 3 − 5 ⋅ −1 1− ( 4 + 3⋅1) <− 3 − 5 ⋅1 1− 1 < −3 + 5 1− 7 < −3 − 5 0<2 ? −6 < −8 x 1− ( 4 + 3⋅ 0 ) <− 3 − 5 ⋅ 0 −3 < −3 x • The solution checks out. {x | x < 0}
  • 144. Example 5 - Solve 1 3(x − 5) ≥ ( 6x + 8 ) 2
  • 145. Example 5 - Solve • Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 ) 2
  • 146. Example 5 - Solve • Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 ) 2 3x − 15 ≥ 3x + 4
  • 147. Example 5 - Solve • Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 ) • Move variables to one side. 2 3x − 15 ≥ 3x + 4
  • 148. Example 5 - Solve • Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 ) • Move variables to one side. 2 3x − 15 ≥ 3x + 4 −3x −3x
  • 149. Example 5 - Solve • Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 ) • Move variables to one side. 2 3x − 15 ≥ 3x + 4 −3x −3x −15 ≥ 4
  • 150. Example 5 - Solve • Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 ) • Move variables to one side. 2 • What happen to x? 3x − 15 ≥ 3x + 4 −3x −3x −15 ≥ 4
  • 151. Example 5 - Solve • Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 ) • Move variables to one side. 2 • What happen to x? 3x − 15 ≥ 3x + 4 • Is the resulting inequality −3x −3x true? −15 ≥ 4
  • 152. Example 5 - Solve • Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 ) • Move variables to one side. 2 • What happen to x? 3x − 15 ≥ 3x + 4 • Is the resulting inequality −3x −3x true? −15 ≥ 4 • Because the result is a false statement, there is no solution to this inequality.
  • 153. Example 5 - Solve • Remove parenthesis. 1 3(x − 5) ≥ ( 6x + 8 ) • Move variables to one side. 2 • What happen to x? 3x − 15 ≥ 3x + 4 • Is the resulting inequality −3x −3x true? −15 ≥ 4 • Because the result is a false statement, there is no No Solution solution to this inequality.
  • 154. Example 6 - Solve 2(5x + 4) ≥ 10x − 3
  • 155. Example 6 - Solve • Remove parenthesis. 2(5x + 4) ≥ 10x − 3
  • 156. Example 6 - Solve • Remove parenthesis. 2(5x + 4) ≥ 10x − 3 10x + 8 ≥ 10x − 3
  • 157. Example 6 - Solve • Remove parenthesis. 2(5x + 4) ≥ 10x − 3 • Move variables to one side. 10x + 8 ≥ 10x − 3
  • 158. Example 6 - Solve • Remove parenthesis. 2(5x + 4) ≥ 10x − 3 • Move variables to one side. 10x + 8 ≥ 10x − 3 −10x −10x
  • 159. Example 6 - Solve • Remove parenthesis. 2(5x + 4) ≥ 10x − 3 • Move variables to one side. 10x + 8 ≥ 10x − 3 −10x −10x 8 ≥ −3
  • 160. Example 6 - Solve • Remove parenthesis. 2(5x + 4) ≥ 10x − 3 • Move variables to one side. • The x disappeared again but is the 10x + 8 ≥ 10x − 3 resulting inequality true? −10x −10x 8 ≥ −3
  • 161. Example 6 - Solve • Remove parenthesis. 2(5x + 4) ≥ 10x − 3 • Move variables to one side. • The x disappeared again but is the 10x + 8 ≥ 10x − 3 resulting inequality true? −10x −10x • Yes! Because the result is a true statement, any number can be 8 ≥ −3 substituted into the original inequality.
  • 162. Example 6 - Solve • Remove parenthesis. 2(5x + 4) ≥ 10x − 3 • Move variables to one side. • The x disappeared again but is the 10x + 8 ≥ 10x − 3 resulting inequality true? −10x −10x • Yes! Because the result is a true statement, any number can be 8 ≥ −3 substituted into the original inequality. All Real Numbers • The graph is the entire number line so the solution is All Real -5 -4 -3 -2 -1 0 1 2 3 4 5 numbers.
  • 163. Example 7 - The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks?
  • 164. Example 7 - The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Remember this problem?
  • 165. Example 7 - The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Remember this problem? • Would $2000 be enough?
  • 166. Example 7 - The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Remember this problem? • Would $2000 be enough? • How about $710?
  • 167. Example 7 - The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Remember this problem? • Would $2000 be enough? • How about $710? • What about $1710?
  • 168. Example 7 - The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Remember this problem? • Would $2000 be enough? • How about $710? • What about $1710? • This situation can be modeled with an Inequality. We would use an Inequality instead of an Equation because an exact amount is not needed. There is a range of acceptable values.
  • 169. (Example 7 - continued) The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Let A = Josh’s account balance before writing the checks.
  • 170. (Example 7 - continued) The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Let A = Josh’s account balance before writing the checks. • The minimum balance is $1000 so this is the least amount allowed in the account.
  • 171. (Example 7 - continued) The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Let A = Josh’s account balance before writing the checks. • The minimum balance is $1000 so this is the least amount allowed in the account. • Checks are deducted from the account balance. What operation shows a deduction?
  • 172. (Example 7 - continued) The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Let A = Josh’s account balance before writing the checks. • The minimum balance is $1000 so this is the least amount allowed in the account. • Checks are deducted from the account balance. What operation shows a deduction? • Subtraction!
  • 173. (Example 7 - continued) The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? • Let A = Josh’s account balance before writing the checks. • The minimum balance is $1000 so this is the least amount allowed in the account. • Checks are deducted from the account balance. What operation shows a deduction? • Subtraction! • $525 and $185 need subtracted from the account balance.
  • 174. (Example 7 - continued) The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? Account balance - (sum of checks written) ≥ Min Balance
  • 175. (Example 7 - continued) The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? Account balance - (sum of checks written) ≥ Min Balance A - (525 + 185) ≥ 1000
  • 176. (Example 7 - continued) The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? Account balance - (sum of checks written) ≥ Min Balance A - (525 + 185) ≥ 1000 A - 710 ≥ 1000
  • 177. (Example 7 - continued) The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? Account balance - (sum of checks written) ≥ Min Balance A - (525 + 185) ≥ 1000 A - 710 ≥ 1000 A - 710 + 710 ≥ 1000 + 710
  • 178. (Example 7 - continued) The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? Account balance - (sum of checks written) ≥ Min Balance A - (525 + 185) ≥ 1000 A - 710 ≥ 1000 A - 710 + 710 ≥ 1000 + 710 A ≥ 1710
  • 179. (Example 7 - continued) The local bank in Mathville requires a minimum balance of $1000. If Josh knows he must write checks for $525 for rent and $185 for utilities, how much should he have in his account before writing the checks? Account balance - (sum of checks written) ≥ Min Balance A - (525 + 185) ≥ 1000 A - 710 ≥ 1000 A - 710 + 710 ≥ 1000 + 710 A ≥ 1710 Josh must have at least $1710 in his checking account before writing the checks.
  • 180. Example 8 - Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover.
  • 181. Example 8 - Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. • Let w = width
  • 182. Example 8 - Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. • Let w = width • length = twice as long as the width = 2w.
  • 183. Example 8 - Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. • Let w = width • length = twice as long as the width = 2w. • Enclosing a space refers to perimeter, the distance around the space.
  • 184. Example 8 - Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. • Let w = width • length = twice as long as the width = 2w. • Enclosing a space refers to perimeter, the distance around the space. • This space is rectangular. Opposite sides have the same measure.
  • 185. Example 8 - Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. • Let w = width • length = twice as long as the width = 2w. • Enclosing a space refers to perimeter, the distance around the space. • This space is rectangular. Opposite sides have the same measure. • Perimeter, P, of a rectangle is P = 2l + 2w, where l is length and w is width.
  • 186. Example 8 - Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. • Let w = width • length = twice as long as the width = 2w. • Enclosing a space refers to perimeter, the distance around the space. • This space is rectangular. Opposite sides have the same measure. • Perimeter, P, of a rectangle is P = 2l + 2w, where l is length and w is width. • Because Jim has 96 feet of fence, this is the maximum perimeter.
  • 187. (Example 8 - continued) Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. Maximum Perimeter ≥ 2l + 2w
  • 188. (Example 8 - continued) Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. Maximum Perimeter ≥ 2l + 2w 96 ≥ 2 * 2w + 2 * w
  • 189. (Example 8 - continued) Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. Maximum Perimeter ≥ 2l + 2w 96 ≥ 2 * 2w + 2 * w 96 ≥ 4w + 2w
  • 190. (Example 8 - continued) Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. Maximum Perimeter ≥ 2l + 2w 96 ≥ 2 * 2w + 2 * w 96 ≥ 4w + 2w 96 ≥ 6w
  • 191. (Example 8 - continued) Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. Maximum Perimeter ≥ 2l + 2w 96 ≥ 2 * 2w + 2 * w 96 ≥ 4w + 2w 96 ≥ 6w 96 ÷ 6 ≥ 6w ÷ 6
  • 192. (Example 8 - continued) Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. Maximum Perimeter ≥ 2l + 2w 96 ≥ 2 * 2w + 2 * w 96 ≥ 4w + 2w 96 ≥ 6w 96 ÷ 6 ≥ 6w ÷ 6 16 ≥ w
  • 193. (Example 8 - continued) Jim has several rectangular spaces that need enclosed by a fence. He has 96 feet of fencing but not sure which space to enclose. All the spaces have lengths twice as long as the width. Write an inequality and solve so Jim can determine the dimensions the fencing will cover. Maximum Perimeter ≥ 2l + 2w 96 ≥ 2 * 2w + 2 * w 96 ≥ 4w + 2w 96 ≥ 6w 96 ÷ 6 ≥ 6w ÷ 6 16 ≥ w Jim can fence a space with a width less than or equal to 16 feet.

Editor's Notes