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Compound Inequalities Notes

L
Lori Rapp

The document discusses compound inequalities, which are statements combining two or more inequalities using AND or OR. AND means the solution must satisfy both inequalities, while OR means it must satisfy at least one. Examples are provided to demonstrate solving and graphing compound inequalities on a number line, including checking solutions in the original inequalities.

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Compound Inequalities
What does compound mean?
What does compound mean? • Compound means to combine 2 or more elements.
What does compound mean? • Compound means to combine 2 or more elements. • In English, this is a word made from 2 words.
What does compound mean? • Compound means to combine 2 or more elements. • In English, this is a word made from 2 words. • Such as lighthouse is a compound word.
What does compound mean? • Compound means to combine 2 or more elements. • In English, this is a word made from 2 words. • Such as lighthouse is a compound word. • In Chemistry, this is 2 or more elements from the periodic table.
What does compound mean? • Compound means to combine 2 or more elements. • In English, this is a word made from 2 words. • Such as lighthouse is a compound word. • In Chemistry, this is 2 or more elements from the periodic table. • Such as water is a compound of the elements hydrogen and oxygen.
Compound Inequality
Compound Inequality • An inequality is a statement such as x > 3.
Compound Inequality • An inequality is a statement such as x > 3. • A Compound Inequality is 2 Inequality statements.
Compound Inequality • An inequality is a statement such as x > 3. • A Compound Inequality is 2 Inequality statements. • Such as:
Compound Inequality • An inequality is a statement such as x > 3. • A Compound Inequality is 2 Inequality statements. • Such as: • x > 3 AND x < 8
Compound Inequality • An inequality is a statement such as x > 3. • A Compound Inequality is 2 Inequality statements. • Such as: • x > 3 AND x < 8 • x < -1 OR x > 0
Compound Inequality (con’t)
Compound Inequality (con’t) • AND means the solution must work in BOTH inequalities
Compound Inequality (con’t) • AND means the solution must work in BOTH inequalities • Example: x > 3 AND x < 8
Compound Inequality (con’t) • AND means the solution must work in BOTH inequalities • Example: x > 3 AND x < 8 • 5 is a solution because 5 > 3 AND 5 <8
Compound Inequality (con’t) • AND means the solution must work in BOTH inequalities • Example: x > 3 AND x < 8 • 5 is a solution because 5 > 3 AND 5 <8 • 2 is not a solution because although 2 < 8, 2 > 3 is not true.
Compound Inequality (con’t) • AND means the solution must work in BOTH inequalities • Example: x > 3 AND x < 8 • 5 is a solution because 5 > 3 AND 5 <8 • 2 is not a solution because although 2 < 8, 2 > 3 is not true. • AND is the intersection (∩) of the inequalities.
Compound Inequality (con’t)
Compound Inequality (con’t) • OR means the solution must work in EITHER inequality.
Compound Inequality (con’t) • OR means the solution must work in EITHER inequality. • Example: x < -1 OR x > 3
Compound Inequality (con’t) • OR means the solution must work in EITHER inequality. • Example: x < -1 OR x > 3 • -5 is a solution because -5 < -1. It doesn’t need to work in x > 3.
Compound Inequality (con’t) • OR means the solution must work in EITHER inequality. • Example: x < -1 OR x > 3 • -5 is a solution because -5 < -1. It doesn’t need to work in x > 3. • 8 is a solution because 8 > 3. It doesn’t need to work in x < -1.
Compound Inequality (con’t) • OR means the solution must work in EITHER inequality. • Example: x < -1 OR x > 3 • -5 is a solution because -5 < -1. It doesn’t need to work in x > 3. • 8 is a solution because 8 > 3. It doesn’t need to work in x < -1. • 0 is not a solution because 0 < -1 is false and 0 > 3 is false. It didn’t work in either inequality.
Compound Inequality (con’t) • OR means the solution must work in EITHER inequality. • Example: x < -1 OR x > 3 • -5 is a solution because -5 < -1. It doesn’t need to work in x > 3. • 8 is a solution because 8 > 3. It doesn’t need to work in x < -1. • 0 is not a solution because 0 < -1 is false and 0 > 3 is false. It didn’t work in either inequality. • OR is the union (∪) of the inequalities.
Example 1 - Solve and Graph: −x + 5 > 7 OR 3x − 7 ≥ 2
Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 normally do.
Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 normally do.
Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 normally do. −x > 2
Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 normally do. −x > 2 −1 −1
Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 normally do. −x > 2 −1 −1 x < −2
Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 +7 +7 normally do. −x > 2 −1 −1 x < −2
Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 +7 +7 normally do. −x > 2 3x ≥ 9 −1 −1 x < −2
Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 +7 +7 normally do. −x > 2 3x ≥ 9 −1 −1 3 3 x < −2
Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 +7 +7 normally do. −x > 2 3x ≥ 9 −1 −1 3 3 x < −2 x≥3
Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 +7 +7 normally do. −x > 2 3x ≥ 9 • Graph each inequality on the same number line. −1 −1 3 3 x < −2 x≥3 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 +7 +7 normally do. −x > 2 3x ≥ 9 • Graph each inequality on the same number line. −1 −1 3 3 x < −2 x≥3 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 +7 +7 normally do. −x > 2 3x ≥ 9 • Graph each inequality on the same number line. −1 −1 3 3 x < −2 x≥3 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 +7 +7 normally do. −x > 2 3x ≥ 9 • Graph each inequality on the same number line. −1 −1 3 3 • Notice the circle is open or closed x < −2 x≥3 based on the inequality being graphed. -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 +7 +7 normally do. −x > 2 3x ≥ 9 • Graph each inequality on the same number line. −1 −1 3 3 • Notice the circle is open or closed x < −2 x≥3 based on the inequality being graphed. • Write the solution. -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 +7 +7 normally do. −x > 2 3x ≥ 9 • Graph each inequality on the same number line. −1 −1 3 3 • Notice the circle is open or closed x < −2 x≥3 based on the inequality being graphed. • Write the solution. -5 -4 -3 -2 -1 0 1 2 3 4 5 {x | x < −2 ∪ x ≥ 3}
Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 +7 +7 normally do. −x > 2 3x ≥ 9 • Graph each inequality on the same number line. −1 −1 3 3 • Notice the circle is open or closed x < −2 x≥3 based on the inequality being graphed. • Write the solution. -5 -4 -3 -2 -1 0 1 2 3 4 5 • This is a Union because a number from either inequality will work in the original problem. Sometimes {x | x < −2 ∪ x ≥ 3} the word or is used in place of ∪.
Example 1 Continued - Check solutions in original Inequalities −x + 5 > 7 OR 3x − 7 ≥ 2 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 1 Continued - Check solutions in original Inequalities −x + 5 > 7 OR 3x − 7 ≥ 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in either inequality. Need the solution to be true in either inequality.
Example 1 Continued - Check solutions in original Inequalities −x + 5 > 7 OR 3x − 7 ≥ 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in either inequality. Need the solution to be true in either inequality. ? − ( −4 ) + 5 > 7 9>7 True
Example 1 Continued - Check solutions in original Inequalities −x + 5 > 7 OR 3x − 7 ≥ 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in either inequality. Need the solution to be true in either inequality. ? ? − ( −4 ) + 5 > 7 3⋅ ( −4 ) − 7 ≥ 2 9>7 −19 ≥ 2 True False
Example 1 Continued - Check solutions in original Inequalities −x + 5 > 7 OR 3x − 7 ≥ 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in either inequality. Need the solution to be true in either inequality. ? ? − ( −4 ) + 5 > 7 3⋅ ( −4 ) − 7 ≥ 2 9>7 −19 ≥ 2 True √ False
Example 1 Continued - Check solutions in original Inequalities −x + 5 > 7 OR 3x − 7 ≥ 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in either inequality. Need the solution to be true in either inequality. ? ? ? − ( −4 ) + 5 > 7 3⋅ ( −4 ) − 7 ≥ 2 −4 + 5 > 7 9>7 −19 ≥ 2 1> 7 True √ False False
Example 1 Continued - Check solutions in original Inequalities −x + 5 > 7 OR 3x − 7 ≥ 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in either inequality. Need the solution to be true in either inequality. ? ? ? ? − ( −4 ) + 5 > 7 3⋅ ( −4 ) − 7 ≥ 2 −4 + 5 > 7 3⋅ 4 − 7 ≥ 2 9>7 −19 ≥ 2 1> 7 5≥2 True √ False False True
Example 1 Continued - Check solutions in original Inequalities −x + 5 > 7 OR 3x − 7 ≥ 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in either inequality. Need the solution to be true in either inequality. ? ? ? ? − ( −4 ) + 5 > 7 3⋅ ( −4 ) − 7 ≥ 2 −4 + 5 > 7 3⋅ 4 − 7 ≥ 2 9>7 −19 ≥ 2 1> 7 5≥2 True √ False False √ True
Example 1 Continued - Check solutions in original Inequalities −x + 5 > 7 OR 3x − 7 ≥ 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in either inequality. Need the solution to be true in either inequality. ? ? ? ? − ( −4 ) + 5 > 7 3⋅ ( −4 ) − 7 ≥ 2 −4 + 5 > 7 3⋅ 4 − 7 ≥ 2 9>7 −19 ≥ 2 1> 7 5≥2 True √ False False √ True ? −0 + 5 > 7 5>7 False
Example 1 Continued - Check solutions in original Inequalities −x + 5 > 7 OR 3x − 7 ≥ 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in either inequality. Need the solution to be true in either inequality. ? ? ? ? − ( −4 ) + 5 > 7 3⋅ ( −4 ) − 7 ≥ 2 −4 + 5 > 7 3⋅ 4 − 7 ≥ 2 9>7 −19 ≥ 2 1> 7 5≥2 True √ False False √ True ? ? −0 + 5 > 7 3⋅ 0 − 7 ≥ 2 5>7 −7 ≥ 2 False False
Example 1 Continued - Check solutions in original Inequalities −x + 5 > 7 OR 3x − 7 ≥ 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in either inequality. Need the solution to be true in either inequality. ? ? ? ? − ( −4 ) + 5 > 7 3⋅ ( −4 ) − 7 ≥ 2 −4 + 5 > 7 3⋅ 4 − 7 ≥ 2 9>7 −19 ≥ 2 1> 7 5≥2 True √ False False √ True ? ? −0 + 5 > 7 3⋅ 0 − 7 ≥ 2 5>7 −7 ≥ 2 False False x
Example 2 - Solve and Graph: x+5<8 AND − x − 3 ≤ −2
Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do.
Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5
Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 x<3
Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 +3 +3 x<3
Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 +3 +3 x<3 −x ≤ 1
Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 +3 +3 x<3 −x ≤ 1 −1 −1
Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 +3 +3 x<3 −x ≤ 1 −1 −1 x ≥ −1
Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 +3 +3 • Graph each inequality on the same x<3 −x ≤ 1 number line. −1 −1 x ≥ −1 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 +3 +3 • Graph each inequality on the same x<3 −x ≤ 1 number line. −1 −1 x ≥ −1 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 +3 +3 • Graph each inequality on the same x<3 −x ≤ 1 number line. −1 −1 x ≥ −1 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 +3 +3 • Graph each inequality on the same x<3 −x ≤ 1 number line. −1 −1 • Because of the word and, the solution must work in both original x ≥ −1 inequalities. -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 +3 +3 • Graph each inequality on the same x<3 −x ≤ 1 number line. −1 −1 • Because of the word and, the solution must work in both original x ≥ −1 inequalities. • The part that overlaps works in both original inequalities. -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 +3 +3 • Graph each inequality on the same x<3 −x ≤ 1 number line. −1 −1 • Because of the word and, the solution must work in both original x ≥ −1 inequalities. • The part that overlaps works in both original inequalities. -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 +3 +3 • Graph each inequality on the same x<3 −x ≤ 1 number line. −1 −1 • Because of the word and, the solution must work in both original x ≥ −1 inequalities. • The part that overlaps works in both original inequalities. -5 -4 -3 -2 -1 0 1 2 3 4 5 • This is an Intersection because a number must work in both original inequalities. Sometimes the word and is used instead of ∩.
Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 +3 +3 • Graph each inequality on the same x<3 −x ≤ 1 number line. −1 −1 • Because of the word and, the solution must work in both original x ≥ −1 inequalities. • The part that overlaps works in both original inequalities. -5 -4 -3 -2 -1 0 1 2 3 4 5 • This is an Intersection because a number must work in both original {x | x < 3∩ x ≥ −1} inequalities. Sometimes the word and is used instead of ∩.
Example 2 Continued- Write AND solutions x+5<8 AND − x − 3 ≤ −2
Example 2 Continued- Write AND solutions • Our last problem gave the x+5<8 AND − x − 3 ≤ −2 following solutions and graph.
Example 2 Continued- Write AND solutions • Our last problem gave the x+5<8 AND − x − 3 ≤ −2 following solutions and graph. x<3 x ≥ −1 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 2 Continued- Write AND solutions • Our last problem gave the x+5<8 AND − x − 3 ≤ −2 following solutions and graph. x<3 x ≥ −1 • The solution was written as an Intersection. -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 2 Continued- Write AND solutions • Our last problem gave the x+5<8 AND − x − 3 ≤ −2 following solutions and graph. x<3 x ≥ −1 • The solution was written as an Intersection. {x | x < 3∩ x ≥ −1} -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 2 Continued- Write AND solutions • Our last problem gave the x+5<8 AND − x − 3 ≤ −2 following solutions and graph. x<3 x ≥ −1 • The solution was written as an Intersection. {x | x < 3∩ x ≥ −1} • The solution can also be written as a combined inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 {x | − 1 ≤ x < 3}
Example 2 Continued- Write AND solutions • Our last problem gave the x+5<8 AND − x − 3 ≤ −2 following solutions and graph. x<3 x ≥ −1 • The solution was written as an Intersection. {x | x < 3∩ x ≥ −1} • The solution can also be written as a combined inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 {x | − 1 ≤ x < 3}
Example 2 Continued- Write AND solutions • Our last problem gave the x+5<8 AND − x − 3 ≤ −2 following solutions and graph. x<3 x ≥ −1 • The solution was written as an Intersection. {x | x < 3∩ x ≥ −1} • The solution can also be written as a combined inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 {x | − 1 ≤ x < 3}
Example 2 Continued- Write AND solutions • Our last problem gave the x+5<8 AND − x − 3 ≤ −2 following solutions and graph. x<3 x ≥ −1 • The solution was written as an Intersection. {x | x < 3∩ x ≥ −1} • The solution can also be written as a combined inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 {x | − 1 ≤ x < 3}
Example 2 Continued- Write AND solutions • Our last problem gave the x+5<8 AND − x − 3 ≤ −2 following solutions and graph. x<3 x ≥ −1 • The solution was written as an Intersection. {x | x < 3∩ x ≥ −1} • The solution can also be written as a combined inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 {x | − 1 ≤ x < 3}
Example 2 Continued- Write AND solutions • Our last problem gave the x+5<8 AND − x − 3 ≤ −2 following solutions and graph. x<3 x ≥ −1 • The solution was written as an Intersection. {x | x < 3∩ x ≥ −1} • The solution can also be written as a combined inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 • The numbers are written from smallest to largest and the inequalities are always either < {x | − 1 ≤ x < 3} or ≤ because the smaller number is always to the left.
Example 2 (Continued) - Check solutions in original Inequality x+5<8 AND − x − 3 ≤ −2 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 2 (Continued) - Check solutions in original Inequality x+5<8 AND − x − 3 ≤ −2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in both inequalities.
Example 2 (Continued) - Check solutions in original Inequality x+5<8 AND − x − 3 ≤ −2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in both inequalities. ? −2 + 5 < 8 3< 8 True
Example 2 (Continued) - Check solutions in original Inequality x+5<8 AND − x − 3 ≤ −2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in both inequalities. ? ? −2 + 5 < 8 −2 − 3≤− 2 3< 8 −5 ≤ −2 True False
Example 2 (Continued) - Check solutions in original Inequality x+5<8 AND − x − 3 ≤ −2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in both inequalities. ? ? −2 + 5 < 8 −2 − 3≤− 2 3< 8 −5 ≤ −2 True x False
Example 2 (Continued) - Check solutions in original Inequality x+5<8 AND − x − 3 ≤ −2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in both inequalities. ? ? ? −2 + 5 < 8 −2 − 3≤− 2 0 + 5<8 3< 8 −5 ≤ −2 5<8 True x False True
Example 2 (Continued) - Check solutions in original Inequality x+5<8 AND − x − 3 ≤ −2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in both inequalities. ? ? ? ? −2 + 5 < 8 −2 − 3≤− 2 0 + 5<8 −0 − 3≤− 2 3< 8 −5 ≤ −2 5<8 −3 < −2 True x False True True
Example 2 (Continued) - Check solutions in original Inequality x+5<8 AND − x − 3 ≤ −2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in both inequalities. ? ? ? ? −2 + 5 < 8 −2 − 3≤− 2 0 + 5<8 −0 − 3≤− 2 3< 8 −5 ≤ −2 5<8 −3 < −2 True x False True √ True
Example 2 (Continued) - Check solutions in original Inequality x+5<8 AND − x − 3 ≤ −2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in both inequalities. ? ? ? ? −2 + 5 < 8 −2 − 3≤− 2 0 + 5<8 −0 − 3≤− 2 3< 8 −5 ≤ −2 5<8 −3 < −2 True x False True √ True ? 5 + 5<8 10 < 8 False
Example 2 (Continued) - Check solutions in original Inequality x+5<8 AND − x − 3 ≤ −2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in both inequalities. ? ? ? ? −2 + 5 < 8 −2 − 3≤− 2 0 + 5<8 −0 − 3≤− 2 3< 8 −5 ≤ −2 5<8 −3 < −2 True x False True √ True ? ? 5 + 5<8 −5 − 3≤− 2 10 < 8 −8 ≤ −2 False True
Example 2 (Continued) - Check solutions in original Inequality x+5<8 AND − x − 3 ≤ −2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in both inequalities. ? ? ? ? −2 + 5 < 8 −2 − 3≤− 2 0 + 5<8 −0 − 3≤− 2 3< 8 −5 ≤ −2 5<8 −3 < −2 True x False True √ True ? ? 5 + 5<8 −5 − 3≤− 2 10 < 8 −8 ≤ −2 False x True
Example 3 - Solve -5 < 2x + 3 ≤ 17
Example 3 - Solve -5 < 2x + 3 ≤ 17 • This is an and compound inequality because the 2x + 3 is between -5 and 17.
Example 3 - Solve -5 < 2x + 3 ≤ 17 • This is an and compound inequality because the 2x + 3 is between -5 and 17. • Notice the small part of both inequality symbol points to the left, just as the smallest numbers on a number line are to the left.
Example 3 - Solve -5 < 2x + 3 ≤ 17 • This is an and compound inequality because the 2x + 3 is between -5 and 17. • Notice the small part of both inequality symbol points to the left, just as the smallest numbers on a number line are to the left. • There are 2 methods for solving.
Example 3 - Solve -5 < 2x + 3 ≤ 17 • This is an and compound inequality because the 2x + 3 is between -5 and 17. • Notice the small part of both inequality symbol points to the left, just as the smallest numbers on a number line are to the left. • There are 2 methods for solving. 1) Split into 2 inequalities and solve each individually.
Example 3 - Solve -5 < 2x + 3 ≤ 17 • This is an and compound inequality because the 2x + 3 is between -5 and 17. • Notice the small part of both inequality symbol points to the left, just as the smallest numbers on a number line are to the left. • There are 2 methods for solving. 1) Split into 2 inequalities and solve each individually. 2) Isolate x in the middle but when you undo operations, keep it balances by doing to the left and the right.
Example 3 - Solve -5 < 2x + 3 ≤ 17 • This is an and compound inequality because the 2x + 3 is between -5 and 17. • Notice the small part of both inequality symbol points to the left, just as the smallest numbers on a number line are to the left. • There are 2 methods for solving. 1) Split into 2 inequalities and solve each individually. 2) Isolate x in the middle but when you undo operations, keep it balances by doing to the left and the right. • It doesn’t matter which method you use, you will get the same answer with both methods.
Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17
Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split
Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split −5 < 2x + 3 2x + 3 ≤ 17
Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split −5 < 2x + 3 2x + 3 ≤ 17 −3 −3
Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split −5 < 2x + 3 2x + 3 ≤ 17 −3 −3 −8 < 2x
Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split −5 < 2x + 3 2x + 3 ≤ 17 −3 −3 −8 < 2x 2 2
Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split −5 < 2x + 3 2x + 3 ≤ 17 −3 −3 −8 < 2x 2 2 −4 < x
Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split −5 < 2x + 3 2x + 3 ≤ 17 −3 −3 −3 −3 −8 < 2x 2 2 −4 < x
Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split −5 < 2x + 3 2x + 3 ≤ 17 −3 −3 −3 −3 −8 < 2x 2x ≤ 14 2 2 −4 < x
Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split −5 < 2x + 3 2x + 3 ≤ 17 −3 −3 −3 −3 −8 < 2x 2x ≤ 14 2 2 2 2 −4 < x
Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split −5 < 2x + 3 2x + 3 ≤ 17 −3 −3 −3 −3 −8 < 2x 2x ≤ 14 2 2 2 2 −4 < x x≤7
Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split Method 2 - Balance −5 < 2x + 3 2x + 3 ≤ 17 −3 −3 −3 −3 −8 < 2x 2x ≤ 14 2 2 2 2 −4 < x x≤7
Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split Method 2 - Balance −5 < 2x + 3 2x + 3 ≤ 17 −5 < 2x + 3 ≤ 17 −3 −3 −3 −3 −8 < 2x 2x ≤ 14 2 2 2 2 −4 < x x≤7
Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split Method 2 - Balance −5 < 2x + 3 2x + 3 ≤ 17 −5 < 2x + 3 ≤ 17 −3 −3 −3 −3 −3 −3 −3 −8 < 2x 2x ≤ 14 2 2 2 2 −4 < x x≤7
Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split Method 2 - Balance −5 < 2x + 3 2x + 3 ≤ 17 −5 < 2x + 3 ≤ 17 −3 −3 −3 −3 −3 −3 −3 −8 < 2x 2x ≤ 14 −8 < 2x ≤ 14 2 2 2 2 −4 < x x≤7
Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split Method 2 - Balance −5 < 2x + 3 2x + 3 ≤ 17 −5 < 2x + 3 ≤ 17 −3 −3 −3 −3 −3 −3 −3 −8 < 2x 2x ≤ 14 −8 < 2x ≤ 14 2 2 2 2 2 2 2 −4 < x x≤7
Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split Method 2 - Balance −5 < 2x + 3 2x + 3 ≤ 17 −5 < 2x + 3 ≤ 17 −3 −3 −3 −3 −3 −3 −3 −8 < 2x 2x ≤ 14 −8 < 2x ≤ 14 2 2 2 2 2 2 2 −4 < x x≤7 −4 < x ≤ 7
Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split Method 2 - Balance −5 < 2x + 3 2x + 3 ≤ 17 −5 < 2x + 3 ≤ 17 −3 −3 −3 −3 −3 −3 −3 −8 < 2x 2x ≤ 14 −8 < 2x ≤ 14 2 2 2 2 2 2 2 −4 < x x≤7 −4 < x ≤ 7 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split Method 2 - Balance −5 < 2x + 3 2x + 3 ≤ 17 −5 < 2x + 3 ≤ 17 −3 −3 −3 −3 −3 −3 −3 −8 < 2x 2x ≤ 14 −8 < 2x ≤ 14 2 2 2 2 2 2 2 −4 < x x≤7 −4 < x ≤ 7 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 {x | −4 < x ≤ 7}
Example 3 - Check solutions in original Inequality −5 < 2x + 3 ≤ 17 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
Example 3 - Check solutions in original Inequality −5 < 2x + 3 ≤ 17 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 • The number must be true in both inequalities.
Example 3 - Check solutions in original Inequality −5 < 2x + 3 ≤ 17 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 • The number must be true in both inequalities. ? ? −5 < 2 ⋅ ( −5 ) + 3≤17 −5 < −7 ≤ 17 −5 < −7 False
Example 3 - Check solutions in original Inequality −5 < 2x + 3 ≤ 17 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 • The number must be true in both inequalities. ? ? −5 < 2 ⋅ ( −5 ) + 3≤17 −5 < −7 ≤ 17 −5 < −7 False x
Example 3 - Check solutions in original Inequality −5 < 2x + 3 ≤ 17 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 • The number must be true in both inequalities. ? ? −5 < 2 ⋅ ( −5 ) + 3≤17 −5 < −7 ≤ 17 ? ? −5 < −7 −5 < 2 ⋅ 0 + 3≤17 −5 < 3 ≤ 17 False x True
Example 3 - Check solutions in original Inequality −5 < 2x + 3 ≤ 17 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 • The number must be true in both inequalities. ? ? −5 < 2 ⋅ ( −5 ) + 3≤17 −5 < −7 ≤ 17 ? ? −5 < −7 −5 < 2 ⋅ 0 + 3≤17 −5 < 3 ≤ 17 False x True √
Example 3 - Check solutions in original Inequality −5 < 2x + 3 ≤ 17 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 • The number must be true in both inequalities. ? ? −5 < 2 ⋅ ( −5 ) + 3≤17 −5 < −7 ≤ 17 ? ? −5 < −7 −5 < 2 ⋅ 0 + 3≤17 ? ? −5 < 3 ≤ 17 False x −5 < 2 ⋅ 8 + 3≤17 True √ −5 < 19 ≤ 17 19 ≤ 17 False
Example 3 - Check solutions in original Inequality −5 < 2x + 3 ≤ 17 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 • The number must be true in both inequalities. ? ? −5 < 2 ⋅ ( −5 ) + 3≤17 −5 < −7 ≤ 17 ? ? −5 < −7 −5 < 2 ⋅ 0 + 3≤17 ? ? −5 < 3 ≤ 17 False x −5 < 2 ⋅ 8 + 3≤17 True √ −5 < 19 ≤ 17 19 ≤ 17 False x
Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0
Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0 • Solve and graph.
Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0 • Solve and graph. +3 +3
Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0 • Solve and graph. +3 +3 2x ≥ 2
Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0 • Solve and graph. +3 +3 2x ≥ 2 2 2
Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0 • Solve and graph. +3 +3 2x ≥ 2 2 2 x ≥1
Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0 • Solve and graph. +3 +3 2x ≥ 2 2 2 x ≥1 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0 • Solve and graph. +3 +3 −3 −3 2x ≥ 2 2 2 x ≥1 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0 • Solve and graph. +3 +3 −3 −3 2x ≥ 2 x ≤ −3 2 2 x ≥1 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0 • Solve and graph. +3 +3 −3 −3 2x ≥ 2 x ≤ −3 2 2 x ≥1 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0 • Solve and graph. +3 +3 −3 −3 • Because of the word and, the 2x ≥ 2 x ≤ −3 solution must work in both 2 2 original inequalities. x ≥1 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0 • Solve and graph. +3 +3 −3 −3 • Because of the word and, the 2x ≥ 2 x ≤ −3 solution must work in both 2 2 original inequalities. x ≥1 • Without checking a solution, we can see there is no -5 -4 -3 -2 -1 0 1 2 3 4 5 overlap in the solutions. Because there is no overlap, no number will work in both inequalities.
Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0 • Solve and graph. +3 +3 −3 −3 • Because of the word and, the 2x ≥ 2 x ≤ −3 solution must work in both 2 2 original inequalities. x ≥1 • Without checking a solution, we can see there is no -5 -4 -3 -2 -1 0 1 2 3 4 5 overlap in the solutions. Because there is no overlap, no number will work in both No Solution inequalities.
Example 5 - 2 − x > −3 OR x − 5 > −2
Example 5 - • Solve and graph. 2 − x > −3 OR x − 5 > −2
Example 5 - • Solve and graph. 2 − x > −3 OR x − 5 > −2 −2 −2
Example 5 - • Solve and graph. 2 − x > −3 OR x − 5 > −2 −2 −2 −x > −5
Example 5 - • Solve and graph. 2 − x > −3 OR x − 5 > −2 −2 −2 −x > −5 −1 −1
Example 5 - • Solve and graph. 2 − x > −3 OR x − 5 > −2 −2 −2 −x > −5 −1 −1 x<5
Example 5 - • Solve and graph. 2 − x > −3 OR x − 5 > −2 −2 −2 −x > −5 −1 −1 x<5 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 5 - • Solve and graph. 2 − x > −3 OR x − 5 > −2 −2 −2 +5 +5 −x > −5 −1 −1 x<5 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 5 - • Solve and graph. 2 − x > −3 OR x − 5 > −2 −2 −2 +5 +5 −x > −5 x>3 −1 −1 x<5 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 5 - • Solve and graph. 2 − x > −3 OR x − 5 > −2 −2 −2 +5 +5 −x > −5 x>3 −1 −1 x<5 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 5 - • Solve and graph. 2 − x > −3 OR x − 5 > −2 −2 −2 +5 +5 • Because of the word or, the solution must work in either −x > −5 x>3 −1 −1 original inequality. x<5 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 5 - • Solve and graph. 2 − x > −3 OR x − 5 > −2 −2 −2 +5 +5 • Because of the word or, the solution must work in either −x > −5 x>3 −1 −1 original inequality. x<5 • Without checking a solution, we can see any number can be chosen. Because the -5 -4 -3 -2 -1 0 1 2 3 4 5 solutions covers the entire number, the solution is All Real numbers.
Example 5 - • Solve and graph. 2 − x > −3 OR x − 5 > −2 −2 −2 +5 +5 • Because of the word or, the solution must work in either −x > −5 x>3 −1 −1 original inequality. x<5 • Without checking a solution, we can see any number can be chosen. Because the -5 -4 -3 -2 -1 0 1 2 3 4 5 solutions covers the entire number, the solution is All All Real Numbers Real numbers.

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Compound Inequalities Notes

  • 3. What does compound mean? • Compound means to combine 2 or more elements.
  • 4. What does compound mean? • Compound means to combine 2 or more elements. • In English, this is a word made from 2 words.
  • 5. What does compound mean? • Compound means to combine 2 or more elements. • In English, this is a word made from 2 words. • Such as lighthouse is a compound word.
  • 6. What does compound mean? • Compound means to combine 2 or more elements. • In English, this is a word made from 2 words. • Such as lighthouse is a compound word. • In Chemistry, this is 2 or more elements from the periodic table.
  • 7. What does compound mean? • Compound means to combine 2 or more elements. • In English, this is a word made from 2 words. • Such as lighthouse is a compound word. • In Chemistry, this is 2 or more elements from the periodic table. • Such as water is a compound of the elements hydrogen and oxygen.
  • 9. Compound Inequality • An inequality is a statement such as x > 3.
  • 10. Compound Inequality • An inequality is a statement such as x > 3. • A Compound Inequality is 2 Inequality statements.
  • 11. Compound Inequality • An inequality is a statement such as x > 3. • A Compound Inequality is 2 Inequality statements. • Such as:
  • 12. Compound Inequality • An inequality is a statement such as x > 3. • A Compound Inequality is 2 Inequality statements. • Such as: • x > 3 AND x < 8
  • 13. Compound Inequality • An inequality is a statement such as x > 3. • A Compound Inequality is 2 Inequality statements. • Such as: • x > 3 AND x < 8 • x < -1 OR x > 0
  • 15. Compound Inequality (con’t) • AND means the solution must work in BOTH inequalities
  • 16. Compound Inequality (con’t) • AND means the solution must work in BOTH inequalities • Example: x > 3 AND x < 8
  • 17. Compound Inequality (con’t) • AND means the solution must work in BOTH inequalities • Example: x > 3 AND x < 8 • 5 is a solution because 5 > 3 AND 5 <8
  • 18. Compound Inequality (con’t) • AND means the solution must work in BOTH inequalities • Example: x > 3 AND x < 8 • 5 is a solution because 5 > 3 AND 5 <8 • 2 is not a solution because although 2 < 8, 2 > 3 is not true.
  • 19. Compound Inequality (con’t) • AND means the solution must work in BOTH inequalities • Example: x > 3 AND x < 8 • 5 is a solution because 5 > 3 AND 5 <8 • 2 is not a solution because although 2 < 8, 2 > 3 is not true. • AND is the intersection (∩) of the inequalities.
  • 21. Compound Inequality (con’t) • OR means the solution must work in EITHER inequality.
  • 22. Compound Inequality (con’t) • OR means the solution must work in EITHER inequality. • Example: x < -1 OR x > 3
  • 23. Compound Inequality (con’t) • OR means the solution must work in EITHER inequality. • Example: x < -1 OR x > 3 • -5 is a solution because -5 < -1. It doesn’t need to work in x > 3.
  • 24. Compound Inequality (con’t) • OR means the solution must work in EITHER inequality. • Example: x < -1 OR x > 3 • -5 is a solution because -5 < -1. It doesn’t need to work in x > 3. • 8 is a solution because 8 > 3. It doesn’t need to work in x < -1.
  • 25. Compound Inequality (con’t) • OR means the solution must work in EITHER inequality. • Example: x < -1 OR x > 3 • -5 is a solution because -5 < -1. It doesn’t need to work in x > 3. • 8 is a solution because 8 > 3. It doesn’t need to work in x < -1. • 0 is not a solution because 0 < -1 is false and 0 > 3 is false. It didn’t work in either inequality.
  • 26. Compound Inequality (con’t) • OR means the solution must work in EITHER inequality. • Example: x < -1 OR x > 3 • -5 is a solution because -5 < -1. It doesn’t need to work in x > 3. • 8 is a solution because 8 > 3. It doesn’t need to work in x < -1. • 0 is not a solution because 0 < -1 is false and 0 > 3 is false. It didn’t work in either inequality. • OR is the union (∪) of the inequalities.
  • 27. Example 1 - Solve and Graph: −x + 5 > 7 OR 3x − 7 ≥ 2
  • 28. Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 normally do.
  • 29. Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 normally do.
  • 30. Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 normally do. −x > 2
  • 31. Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 normally do. −x > 2 −1 −1
  • 32. Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 normally do. −x > 2 −1 −1 x < −2
  • 33. Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 +7 +7 normally do. −x > 2 −1 −1 x < −2
  • 34. Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 +7 +7 normally do. −x > 2 3x ≥ 9 −1 −1 x < −2
  • 35. Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 +7 +7 normally do. −x > 2 3x ≥ 9 −1 −1 3 3 x < −2
  • 36. Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 +7 +7 normally do. −x > 2 3x ≥ 9 −1 −1 3 3 x < −2 x≥3
  • 37. Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 +7 +7 normally do. −x > 2 3x ≥ 9 • Graph each inequality on the same number line. −1 −1 3 3 x < −2 x≥3 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 38. Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 +7 +7 normally do. −x > 2 3x ≥ 9 • Graph each inequality on the same number line. −1 −1 3 3 x < −2 x≥3 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 39. Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 +7 +7 normally do. −x > 2 3x ≥ 9 • Graph each inequality on the same number line. −1 −1 3 3 x < −2 x≥3 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 40. Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 +7 +7 normally do. −x > 2 3x ≥ 9 • Graph each inequality on the same number line. −1 −1 3 3 • Notice the circle is open or closed x < −2 x≥3 based on the inequality being graphed. -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 41. Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 +7 +7 normally do. −x > 2 3x ≥ 9 • Graph each inequality on the same number line. −1 −1 3 3 • Notice the circle is open or closed x < −2 x≥3 based on the inequality being graphed. • Write the solution. -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 42. Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 +7 +7 normally do. −x > 2 3x ≥ 9 • Graph each inequality on the same number line. −1 −1 3 3 • Notice the circle is open or closed x < −2 x≥3 based on the inequality being graphed. • Write the solution. -5 -4 -3 -2 -1 0 1 2 3 4 5 {x | x < −2 ∪ x ≥ 3}
  • 43. Example 1 - Solve and Graph: • Solve each inequality as you −x + 5 > 7 OR 3x − 7 ≥ 2 −5 −5 +7 +7 normally do. −x > 2 3x ≥ 9 • Graph each inequality on the same number line. −1 −1 3 3 • Notice the circle is open or closed x < −2 x≥3 based on the inequality being graphed. • Write the solution. -5 -4 -3 -2 -1 0 1 2 3 4 5 • This is a Union because a number from either inequality will work in the original problem. Sometimes {x | x < −2 ∪ x ≥ 3} the word or is used in place of ∪.
  • 44. Example 1 Continued - Check solutions in original Inequalities −x + 5 > 7 OR 3x − 7 ≥ 2 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 45. Example 1 Continued - Check solutions in original Inequalities −x + 5 > 7 OR 3x − 7 ≥ 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in either inequality. Need the solution to be true in either inequality.
  • 46. Example 1 Continued - Check solutions in original Inequalities −x + 5 > 7 OR 3x − 7 ≥ 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in either inequality. Need the solution to be true in either inequality. ? − ( −4 ) + 5 > 7 9>7 True
  • 47. Example 1 Continued - Check solutions in original Inequalities −x + 5 > 7 OR 3x − 7 ≥ 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in either inequality. Need the solution to be true in either inequality. ? ? − ( −4 ) + 5 > 7 3⋅ ( −4 ) − 7 ≥ 2 9>7 −19 ≥ 2 True False
  • 48. Example 1 Continued - Check solutions in original Inequalities −x + 5 > 7 OR 3x − 7 ≥ 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in either inequality. Need the solution to be true in either inequality. ? ? − ( −4 ) + 5 > 7 3⋅ ( −4 ) − 7 ≥ 2 9>7 −19 ≥ 2 True √ False
  • 49. Example 1 Continued - Check solutions in original Inequalities −x + 5 > 7 OR 3x − 7 ≥ 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in either inequality. Need the solution to be true in either inequality. ? ? ? − ( −4 ) + 5 > 7 3⋅ ( −4 ) − 7 ≥ 2 −4 + 5 > 7 9>7 −19 ≥ 2 1> 7 True √ False False
  • 50. Example 1 Continued - Check solutions in original Inequalities −x + 5 > 7 OR 3x − 7 ≥ 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in either inequality. Need the solution to be true in either inequality. ? ? ? ? − ( −4 ) + 5 > 7 3⋅ ( −4 ) − 7 ≥ 2 −4 + 5 > 7 3⋅ 4 − 7 ≥ 2 9>7 −19 ≥ 2 1> 7 5≥2 True √ False False True
  • 51. Example 1 Continued - Check solutions in original Inequalities −x + 5 > 7 OR 3x − 7 ≥ 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in either inequality. Need the solution to be true in either inequality. ? ? ? ? − ( −4 ) + 5 > 7 3⋅ ( −4 ) − 7 ≥ 2 −4 + 5 > 7 3⋅ 4 − 7 ≥ 2 9>7 −19 ≥ 2 1> 7 5≥2 True √ False False √ True
  • 52. Example 1 Continued - Check solutions in original Inequalities −x + 5 > 7 OR 3x − 7 ≥ 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in either inequality. Need the solution to be true in either inequality. ? ? ? ? − ( −4 ) + 5 > 7 3⋅ ( −4 ) − 7 ≥ 2 −4 + 5 > 7 3⋅ 4 − 7 ≥ 2 9>7 −19 ≥ 2 1> 7 5≥2 True √ False False √ True ? −0 + 5 > 7 5>7 False
  • 53. Example 1 Continued - Check solutions in original Inequalities −x + 5 > 7 OR 3x − 7 ≥ 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in either inequality. Need the solution to be true in either inequality. ? ? ? ? − ( −4 ) + 5 > 7 3⋅ ( −4 ) − 7 ≥ 2 −4 + 5 > 7 3⋅ 4 − 7 ≥ 2 9>7 −19 ≥ 2 1> 7 5≥2 True √ False False √ True ? ? −0 + 5 > 7 3⋅ 0 − 7 ≥ 2 5>7 −7 ≥ 2 False False
  • 54. Example 1 Continued - Check solutions in original Inequalities −x + 5 > 7 OR 3x − 7 ≥ 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in either inequality. Need the solution to be true in either inequality. ? ? ? ? − ( −4 ) + 5 > 7 3⋅ ( −4 ) − 7 ≥ 2 −4 + 5 > 7 3⋅ 4 − 7 ≥ 2 9>7 −19 ≥ 2 1> 7 5≥2 True √ False False √ True ? ? −0 + 5 > 7 3⋅ 0 − 7 ≥ 2 5>7 −7 ≥ 2 False False x
  • 55. Example 2 - Solve and Graph: x+5<8 AND − x − 3 ≤ −2
  • 56. Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do.
  • 57. Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5
  • 58. Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 x<3
  • 59. Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 +3 +3 x<3
  • 60. Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 +3 +3 x<3 −x ≤ 1
  • 61. Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 +3 +3 x<3 −x ≤ 1 −1 −1
  • 62. Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 +3 +3 x<3 −x ≤ 1 −1 −1 x ≥ −1
  • 63. Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 +3 +3 • Graph each inequality on the same x<3 −x ≤ 1 number line. −1 −1 x ≥ −1 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 64. Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 +3 +3 • Graph each inequality on the same x<3 −x ≤ 1 number line. −1 −1 x ≥ −1 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 65. Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 +3 +3 • Graph each inequality on the same x<3 −x ≤ 1 number line. −1 −1 x ≥ −1 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 66. Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 +3 +3 • Graph each inequality on the same x<3 −x ≤ 1 number line. −1 −1 • Because of the word and, the solution must work in both original x ≥ −1 inequalities. -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 67. Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 +3 +3 • Graph each inequality on the same x<3 −x ≤ 1 number line. −1 −1 • Because of the word and, the solution must work in both original x ≥ −1 inequalities. • The part that overlaps works in both original inequalities. -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 68. Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 +3 +3 • Graph each inequality on the same x<3 −x ≤ 1 number line. −1 −1 • Because of the word and, the solution must work in both original x ≥ −1 inequalities. • The part that overlaps works in both original inequalities. -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 69. Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 +3 +3 • Graph each inequality on the same x<3 −x ≤ 1 number line. −1 −1 • Because of the word and, the solution must work in both original x ≥ −1 inequalities. • The part that overlaps works in both original inequalities. -5 -4 -3 -2 -1 0 1 2 3 4 5 • This is an Intersection because a number must work in both original inequalities. Sometimes the word and is used instead of ∩.
  • 70. Example 2 - Solve and Graph: • Solve each inequality as you x+5<8 AND − x − 3 ≤ −2 normally do. −5 −5 +3 +3 • Graph each inequality on the same x<3 −x ≤ 1 number line. −1 −1 • Because of the word and, the solution must work in both original x ≥ −1 inequalities. • The part that overlaps works in both original inequalities. -5 -4 -3 -2 -1 0 1 2 3 4 5 • This is an Intersection because a number must work in both original {x | x < 3∩ x ≥ −1} inequalities. Sometimes the word and is used instead of ∩.
  • 71. Example 2 Continued- Write AND solutions x+5<8 AND − x − 3 ≤ −2
  • 72. Example 2 Continued- Write AND solutions • Our last problem gave the x+5<8 AND − x − 3 ≤ −2 following solutions and graph.
  • 73. Example 2 Continued- Write AND solutions • Our last problem gave the x+5<8 AND − x − 3 ≤ −2 following solutions and graph. x<3 x ≥ −1 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 74. Example 2 Continued- Write AND solutions • Our last problem gave the x+5<8 AND − x − 3 ≤ −2 following solutions and graph. x<3 x ≥ −1 • The solution was written as an Intersection. -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 75. Example 2 Continued- Write AND solutions • Our last problem gave the x+5<8 AND − x − 3 ≤ −2 following solutions and graph. x<3 x ≥ −1 • The solution was written as an Intersection. {x | x < 3∩ x ≥ −1} -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 76. Example 2 Continued- Write AND solutions • Our last problem gave the x+5<8 AND − x − 3 ≤ −2 following solutions and graph. x<3 x ≥ −1 • The solution was written as an Intersection. {x | x < 3∩ x ≥ −1} • The solution can also be written as a combined inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 {x | − 1 ≤ x < 3}
  • 77. Example 2 Continued- Write AND solutions • Our last problem gave the x+5<8 AND − x − 3 ≤ −2 following solutions and graph. x<3 x ≥ −1 • The solution was written as an Intersection. {x | x < 3∩ x ≥ −1} • The solution can also be written as a combined inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 {x | − 1 ≤ x < 3}
  • 78. Example 2 Continued- Write AND solutions • Our last problem gave the x+5<8 AND − x − 3 ≤ −2 following solutions and graph. x<3 x ≥ −1 • The solution was written as an Intersection. {x | x < 3∩ x ≥ −1} • The solution can also be written as a combined inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 {x | − 1 ≤ x < 3}
  • 79. Example 2 Continued- Write AND solutions • Our last problem gave the x+5<8 AND − x − 3 ≤ −2 following solutions and graph. x<3 x ≥ −1 • The solution was written as an Intersection. {x | x < 3∩ x ≥ −1} • The solution can also be written as a combined inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 {x | − 1 ≤ x < 3}
  • 80. Example 2 Continued- Write AND solutions • Our last problem gave the x+5<8 AND − x − 3 ≤ −2 following solutions and graph. x<3 x ≥ −1 • The solution was written as an Intersection. {x | x < 3∩ x ≥ −1} • The solution can also be written as a combined inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 {x | − 1 ≤ x < 3}
  • 81. Example 2 Continued- Write AND solutions • Our last problem gave the x+5<8 AND − x − 3 ≤ −2 following solutions and graph. x<3 x ≥ −1 • The solution was written as an Intersection. {x | x < 3∩ x ≥ −1} • The solution can also be written as a combined inequality. -5 -4 -3 -2 -1 0 1 2 3 4 5 • The numbers are written from smallest to largest and the inequalities are always either < {x | − 1 ≤ x < 3} or ≤ because the smaller number is always to the left.
  • 82. Example 2 (Continued) - Check solutions in original Inequality x+5<8 AND − x − 3 ≤ −2 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 83. Example 2 (Continued) - Check solutions in original Inequality x+5<8 AND − x − 3 ≤ −2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in both inequalities.
  • 84. Example 2 (Continued) - Check solutions in original Inequality x+5<8 AND − x − 3 ≤ −2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in both inequalities. ? −2 + 5 < 8 3< 8 True
  • 85. Example 2 (Continued) - Check solutions in original Inequality x+5<8 AND − x − 3 ≤ −2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in both inequalities. ? ? −2 + 5 < 8 −2 − 3≤− 2 3< 8 −5 ≤ −2 True False
  • 86. Example 2 (Continued) - Check solutions in original Inequality x+5<8 AND − x − 3 ≤ −2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in both inequalities. ? ? −2 + 5 < 8 −2 − 3≤− 2 3< 8 −5 ≤ −2 True x False
  • 87. Example 2 (Continued) - Check solutions in original Inequality x+5<8 AND − x − 3 ≤ −2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in both inequalities. ? ? ? −2 + 5 < 8 −2 − 3≤− 2 0 + 5<8 3< 8 −5 ≤ −2 5<8 True x False True
  • 88. Example 2 (Continued) - Check solutions in original Inequality x+5<8 AND − x − 3 ≤ −2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in both inequalities. ? ? ? ? −2 + 5 < 8 −2 − 3≤− 2 0 + 5<8 −0 − 3≤− 2 3< 8 −5 ≤ −2 5<8 −3 < −2 True x False True True
  • 89. Example 2 (Continued) - Check solutions in original Inequality x+5<8 AND − x − 3 ≤ −2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in both inequalities. ? ? ? ? −2 + 5 < 8 −2 − 3≤− 2 0 + 5<8 −0 − 3≤− 2 3< 8 −5 ≤ −2 5<8 −3 < −2 True x False True √ True
  • 90. Example 2 (Continued) - Check solutions in original Inequality x+5<8 AND − x − 3 ≤ −2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in both inequalities. ? ? ? ? −2 + 5 < 8 −2 − 3≤− 2 0 + 5<8 −0 − 3≤− 2 3< 8 −5 ≤ −2 5<8 −3 < −2 True x False True √ True ? 5 + 5<8 10 < 8 False
  • 91. Example 2 (Continued) - Check solutions in original Inequality x+5<8 AND − x − 3 ≤ −2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in both inequalities. ? ? ? ? −2 + 5 < 8 −2 − 3≤− 2 0 + 5<8 −0 − 3≤− 2 3< 8 −5 ≤ −2 5<8 −3 < −2 True x False True √ True ? ? 5 + 5<8 −5 − 3≤− 2 10 < 8 −8 ≤ −2 False True
  • 92. Example 2 (Continued) - Check solutions in original Inequality x+5<8 AND − x − 3 ≤ −2 -5 -4 -3 -2 -1 0 1 2 3 4 5 • The number must be true in both inequalities. ? ? ? ? −2 + 5 < 8 −2 − 3≤− 2 0 + 5<8 −0 − 3≤− 2 3< 8 −5 ≤ −2 5<8 −3 < −2 True x False True √ True ? ? 5 + 5<8 −5 − 3≤− 2 10 < 8 −8 ≤ −2 False x True
  • 93. Example 3 - Solve -5 < 2x + 3 ≤ 17
  • 94. Example 3 - Solve -5 < 2x + 3 ≤ 17 • This is an and compound inequality because the 2x + 3 is between -5 and 17.
  • 95. Example 3 - Solve -5 < 2x + 3 ≤ 17 • This is an and compound inequality because the 2x + 3 is between -5 and 17. • Notice the small part of both inequality symbol points to the left, just as the smallest numbers on a number line are to the left.
  • 96. Example 3 - Solve -5 < 2x + 3 ≤ 17 • This is an and compound inequality because the 2x + 3 is between -5 and 17. • Notice the small part of both inequality symbol points to the left, just as the smallest numbers on a number line are to the left. • There are 2 methods for solving.
  • 97. Example 3 - Solve -5 < 2x + 3 ≤ 17 • This is an and compound inequality because the 2x + 3 is between -5 and 17. • Notice the small part of both inequality symbol points to the left, just as the smallest numbers on a number line are to the left. • There are 2 methods for solving. 1) Split into 2 inequalities and solve each individually.
  • 98. Example 3 - Solve -5 < 2x + 3 ≤ 17 • This is an and compound inequality because the 2x + 3 is between -5 and 17. • Notice the small part of both inequality symbol points to the left, just as the smallest numbers on a number line are to the left. • There are 2 methods for solving. 1) Split into 2 inequalities and solve each individually. 2) Isolate x in the middle but when you undo operations, keep it balances by doing to the left and the right.
  • 99. Example 3 - Solve -5 < 2x + 3 ≤ 17 • This is an and compound inequality because the 2x + 3 is between -5 and 17. • Notice the small part of both inequality symbol points to the left, just as the smallest numbers on a number line are to the left. • There are 2 methods for solving. 1) Split into 2 inequalities and solve each individually. 2) Isolate x in the middle but when you undo operations, keep it balances by doing to the left and the right. • It doesn’t matter which method you use, you will get the same answer with both methods.
  • 100. Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17
  • 101. Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split
  • 102. Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split −5 < 2x + 3 2x + 3 ≤ 17
  • 103. Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split −5 < 2x + 3 2x + 3 ≤ 17 −3 −3
  • 104. Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split −5 < 2x + 3 2x + 3 ≤ 17 −3 −3 −8 < 2x
  • 105. Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split −5 < 2x + 3 2x + 3 ≤ 17 −3 −3 −8 < 2x 2 2
  • 106. Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split −5 < 2x + 3 2x + 3 ≤ 17 −3 −3 −8 < 2x 2 2 −4 < x
  • 107. Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split −5 < 2x + 3 2x + 3 ≤ 17 −3 −3 −3 −3 −8 < 2x 2 2 −4 < x
  • 108. Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split −5 < 2x + 3 2x + 3 ≤ 17 −3 −3 −3 −3 −8 < 2x 2x ≤ 14 2 2 −4 < x
  • 109. Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split −5 < 2x + 3 2x + 3 ≤ 17 −3 −3 −3 −3 −8 < 2x 2x ≤ 14 2 2 2 2 −4 < x
  • 110. Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split −5 < 2x + 3 2x + 3 ≤ 17 −3 −3 −3 −3 −8 < 2x 2x ≤ 14 2 2 2 2 −4 < x x≤7
  • 111. Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split Method 2 - Balance −5 < 2x + 3 2x + 3 ≤ 17 −3 −3 −3 −3 −8 < 2x 2x ≤ 14 2 2 2 2 −4 < x x≤7
  • 112. Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split Method 2 - Balance −5 < 2x + 3 2x + 3 ≤ 17 −5 < 2x + 3 ≤ 17 −3 −3 −3 −3 −8 < 2x 2x ≤ 14 2 2 2 2 −4 < x x≤7
  • 113. Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split Method 2 - Balance −5 < 2x + 3 2x + 3 ≤ 17 −5 < 2x + 3 ≤ 17 −3 −3 −3 −3 −3 −3 −3 −8 < 2x 2x ≤ 14 2 2 2 2 −4 < x x≤7
  • 114. Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split Method 2 - Balance −5 < 2x + 3 2x + 3 ≤ 17 −5 < 2x + 3 ≤ 17 −3 −3 −3 −3 −3 −3 −3 −8 < 2x 2x ≤ 14 −8 < 2x ≤ 14 2 2 2 2 −4 < x x≤7
  • 115. Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split Method 2 - Balance −5 < 2x + 3 2x + 3 ≤ 17 −5 < 2x + 3 ≤ 17 −3 −3 −3 −3 −3 −3 −3 −8 < 2x 2x ≤ 14 −8 < 2x ≤ 14 2 2 2 2 2 2 2 −4 < x x≤7
  • 116. Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split Method 2 - Balance −5 < 2x + 3 2x + 3 ≤ 17 −5 < 2x + 3 ≤ 17 −3 −3 −3 −3 −3 −3 −3 −8 < 2x 2x ≤ 14 −8 < 2x ≤ 14 2 2 2 2 2 2 2 −4 < x x≤7 −4 < x ≤ 7
  • 117. Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split Method 2 - Balance −5 < 2x + 3 2x + 3 ≤ 17 −5 < 2x + 3 ≤ 17 −3 −3 −3 −3 −3 −3 −3 −8 < 2x 2x ≤ 14 −8 < 2x ≤ 14 2 2 2 2 2 2 2 −4 < x x≤7 −4 < x ≤ 7 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
  • 118. Example 3 (Continued) - Solve and Graph: −5 < 2x + 3 ≤ 17 Method 1 - Split Method 2 - Balance −5 < 2x + 3 2x + 3 ≤ 17 −5 < 2x + 3 ≤ 17 −3 −3 −3 −3 −3 −3 −3 −8 < 2x 2x ≤ 14 −8 < 2x ≤ 14 2 2 2 2 2 2 2 −4 < x x≤7 −4 < x ≤ 7 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 {x | −4 < x ≤ 7}
  • 119. Example 3 - Check solutions in original Inequality −5 < 2x + 3 ≤ 17 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
  • 120. Example 3 - Check solutions in original Inequality −5 < 2x + 3 ≤ 17 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 • The number must be true in both inequalities.
  • 121. Example 3 - Check solutions in original Inequality −5 < 2x + 3 ≤ 17 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 • The number must be true in both inequalities. ? ? −5 < 2 ⋅ ( −5 ) + 3≤17 −5 < −7 ≤ 17 −5 < −7 False
  • 122. Example 3 - Check solutions in original Inequality −5 < 2x + 3 ≤ 17 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 • The number must be true in both inequalities. ? ? −5 < 2 ⋅ ( −5 ) + 3≤17 −5 < −7 ≤ 17 −5 < −7 False x
  • 123. Example 3 - Check solutions in original Inequality −5 < 2x + 3 ≤ 17 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 • The number must be true in both inequalities. ? ? −5 < 2 ⋅ ( −5 ) + 3≤17 −5 < −7 ≤ 17 ? ? −5 < −7 −5 < 2 ⋅ 0 + 3≤17 −5 < 3 ≤ 17 False x True
  • 124. Example 3 - Check solutions in original Inequality −5 < 2x + 3 ≤ 17 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 • The number must be true in both inequalities. ? ? −5 < 2 ⋅ ( −5 ) + 3≤17 −5 < −7 ≤ 17 ? ? −5 < −7 −5 < 2 ⋅ 0 + 3≤17 −5 < 3 ≤ 17 False x True √
  • 125. Example 3 - Check solutions in original Inequality −5 < 2x + 3 ≤ 17 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 • The number must be true in both inequalities. ? ? −5 < 2 ⋅ ( −5 ) + 3≤17 −5 < −7 ≤ 17 ? ? −5 < −7 −5 < 2 ⋅ 0 + 3≤17 ? ? −5 < 3 ≤ 17 False x −5 < 2 ⋅ 8 + 3≤17 True √ −5 < 19 ≤ 17 19 ≤ 17 False
  • 126. Example 3 - Check solutions in original Inequality −5 < 2x + 3 ≤ 17 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 • The number must be true in both inequalities. ? ? −5 < 2 ⋅ ( −5 ) + 3≤17 −5 < −7 ≤ 17 ? ? −5 < −7 −5 < 2 ⋅ 0 + 3≤17 ? ? −5 < 3 ≤ 17 False x −5 < 2 ⋅ 8 + 3≤17 True √ −5 < 19 ≤ 17 19 ≤ 17 False x
  • 127. Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0
  • 128. Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0 • Solve and graph.
  • 129. Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0 • Solve and graph. +3 +3
  • 130. Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0 • Solve and graph. +3 +3 2x ≥ 2
  • 131. Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0 • Solve and graph. +3 +3 2x ≥ 2 2 2
  • 132. Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0 • Solve and graph. +3 +3 2x ≥ 2 2 2 x ≥1
  • 133. Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0 • Solve and graph. +3 +3 2x ≥ 2 2 2 x ≥1 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 134. Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0 • Solve and graph. +3 +3 −3 −3 2x ≥ 2 2 2 x ≥1 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 135. Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0 • Solve and graph. +3 +3 −3 −3 2x ≥ 2 x ≤ −3 2 2 x ≥1 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 136. Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0 • Solve and graph. +3 +3 −3 −3 2x ≥ 2 x ≤ −3 2 2 x ≥1 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 137. Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0 • Solve and graph. +3 +3 −3 −3 • Because of the word and, the 2x ≥ 2 x ≤ −3 solution must work in both 2 2 original inequalities. x ≥1 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 138. Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0 • Solve and graph. +3 +3 −3 −3 • Because of the word and, the 2x ≥ 2 x ≤ −3 solution must work in both 2 2 original inequalities. x ≥1 • Without checking a solution, we can see there is no -5 -4 -3 -2 -1 0 1 2 3 4 5 overlap in the solutions. Because there is no overlap, no number will work in both inequalities.
  • 139. Example 4 - 2x − 3 ≥ −1 AND x + 3 ≤ 0 • Solve and graph. +3 +3 −3 −3 • Because of the word and, the 2x ≥ 2 x ≤ −3 solution must work in both 2 2 original inequalities. x ≥1 • Without checking a solution, we can see there is no -5 -4 -3 -2 -1 0 1 2 3 4 5 overlap in the solutions. Because there is no overlap, no number will work in both No Solution inequalities.
  • 140. Example 5 - 2 − x > −3 OR x − 5 > −2
  • 141. Example 5 - • Solve and graph. 2 − x > −3 OR x − 5 > −2
  • 142. Example 5 - • Solve and graph. 2 − x > −3 OR x − 5 > −2 −2 −2
  • 143. Example 5 - • Solve and graph. 2 − x > −3 OR x − 5 > −2 −2 −2 −x > −5
  • 144. Example 5 - • Solve and graph. 2 − x > −3 OR x − 5 > −2 −2 −2 −x > −5 −1 −1
  • 145. Example 5 - • Solve and graph. 2 − x > −3 OR x − 5 > −2 −2 −2 −x > −5 −1 −1 x<5
  • 146. Example 5 - • Solve and graph. 2 − x > −3 OR x − 5 > −2 −2 −2 −x > −5 −1 −1 x<5 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 147. Example 5 - • Solve and graph. 2 − x > −3 OR x − 5 > −2 −2 −2 +5 +5 −x > −5 −1 −1 x<5 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 148. Example 5 - • Solve and graph. 2 − x > −3 OR x − 5 > −2 −2 −2 +5 +5 −x > −5 x>3 −1 −1 x<5 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 149. Example 5 - • Solve and graph. 2 − x > −3 OR x − 5 > −2 −2 −2 +5 +5 −x > −5 x>3 −1 −1 x<5 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 150. Example 5 - • Solve and graph. 2 − x > −3 OR x − 5 > −2 −2 −2 +5 +5 • Because of the word or, the solution must work in either −x > −5 x>3 −1 −1 original inequality. x<5 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 151. Example 5 - • Solve and graph. 2 − x > −3 OR x − 5 > −2 −2 −2 +5 +5 • Because of the word or, the solution must work in either −x > −5 x>3 −1 −1 original inequality. x<5 • Without checking a solution, we can see any number can be chosen. Because the -5 -4 -3 -2 -1 0 1 2 3 4 5 solutions covers the entire number, the solution is All Real numbers.
  • 152. Example 5 - • Solve and graph. 2 − x > −3 OR x − 5 > −2 −2 −2 +5 +5 • Because of the word or, the solution must work in either −x > −5 x>3 −1 −1 original inequality. x<5 • Without checking a solution, we can see any number can be chosen. Because the -5 -4 -3 -2 -1 0 1 2 3 4 5 solutions covers the entire number, the solution is All All Real Numbers Real numbers.

Editor's Notes