Algebra 1. 9.7 Lesson. Absolute Value
- 1. Group Quiz Solve each equation. Bellringer: Finish this group quiz from yesterday. 1. 15 = | x | 2. 2| x – 7| = 14 3. | x + 1|– 9 = –9 4. |5 + x | – 3 = –2 5. 7 + | x – 8| = 6 – 15, 15 0, 14 – 1 – 6, –4 no solution
- 2. 2-4 Agenda Holt Algebra 1 Bellringer – Group Quiz Solving Absolute-Value Equations Book Problems Solving in words activity Worksheet
- 3. Recall that the absolute-value of a number is that number’s distance from zero on a number line. For example, |–5| = 5 and |5| = 5. 5 4 3 2 0 1 2 3 4 5 6 1 6 To write this statement using algebra, you would write | x | = 5. This equation asks, “What values of x have an absolute value of 5?” The solutions are 5 and –5. Notice this equation has two solutions. 5 units 5 units
- 5. Additional Example 1A: Solving Absolute-Value Equations Solve the equation. | x | = 12 | x | = 12 The solutions are {12, –12}. Think: What numbers are 12 units from 0? Rewrite the equation as two cases. Case 1 x = 12 Case 2 x = –12 12 units 12 units 10 8 6 4 0 2 4 6 8 10 12 2 12 • • •
- 6. Additional Example 1A: Solving Absolute-Value Equations Solve the equation. | x | = 9 | x | = 9 The solutions are {9, –9}. Think: What numbers are 9 units from 0? Rewrite the equation as two cases. Case 1 x = 9 Case 2 x = –9 9 units 9 units 10 8 6 4 0 2 4 6 8 10 12 2 12 • • •
- 7. Solve the equation. Check It Out! Example 1b 8 =| x 2.5| Think: What numbers are 8 units from 0? Case 1 8 = x 2.5 Case 2 8 = x 2.5 Rewrite the equations as two cases. The solutions are {10.5, –5.5}. 8 =| x 2.5| Since 2.5 is subtracted from x add 2.5 to both sides of each equation. +2.5 +2.5 10.5 = x +2.5 +2.5 5.5 = x
- 8. Solve the equation. Check It Out! Example 1b 7 =| x 4| Case 1 7 = x 4 Case 2 7 = x 4 The solutions are {11, –3}. 7 =| x 4| +4 +4 11 = x +4 +4 -3 = x
- 9. Solve the equation. Check It Out! Example 1a | x| – 3 = 4 Case 1 x = 7 Case 2 x = –7 The solutions are {7, –7}. Since 3 is subtracted from |x|, add 3 to both sides. Think: What numbers are 7 units from 0? Rewrite the equation as two cases. | x| – 3 = 4 + 3 +3 | x | = 7
- 10. Additional Example 1B: Solving Absolute-Value Equations 3| x + 7| = 24 | x + 7| = 8 The solutions are {1, –15}. Since |x + 7| is multiplied by 3, divide both sides by 3 to undo the multiplication. Think: What numbers are 8 units from 0? Rewrite the equations as two cases. Since 7 is added to x subtract 7 from both sides of each equation. Solve the equation. Case 1 x + 7 = 8 Case 2 x + 7 = –8 – 7 –7 – 7 – 7 x = 1 x = –15
- 11. Check It Out! Example 2b Solve the equation. 6 + | x 4| = 6 Since –6 is added to |x 4|, add 6 to both sides. There is only one case. Since 4 is subtracted from x, add 4 to both sides to undo the addition. 6 + | x 4| = 6 +6 +6 | x 4 | = 0 x 4 = 0 + 4 +4 x = 4
- 12. 2 nd Problem Solve the equation. 7 + | x 3| = -2 x 4 = 5 7 + | x 4| = 2 +7 +7 | x 4 | = 5 x 4 = -5 + 4 +4 x = -1 x 4 = 5 + 4 +4 x = 9
- 13. Additional Example 2B: Special Cases of Absolute-Value Equations Solve the equation. 3 + | x + 4| = 0 Absolute value cannot be negative. This equation has no solution. 3 + | x + 4| = 0 3 3 | x + 4| = 3
- 14. Remember! Absolute value must be nonnegative because it represents a distance.
- 15. Check It Out! Example 2a Solve the equation. 2 |2 x 5| = 7 Since 2 is added to –|2x – 5|, subtract 2 from both sides to undo the addition. Absolute value cannot be negative. |2 x 5| = 5 This equation has no solution. Since |2x – 5| is multiplied by negative 1, divide both sides by negative 1. 2 |2 x 5| = 7 2 2 |2 x 5| = 5
- 16. Word Problems Troy’s car can go 24 miles on one gallon of gasoline. However, his gas mileage can vary from this value by 2 miles per gallon depending on where he drives. Write an absolute-value equation that you can use to find the minimum and maximum gas mileage. │ x – 24 │= 2 Solve the equation to find the minimum and maximum gas mileage. x – 24 = 2 and x – 24 = -2 + 24 +24 + 24 + 24 x = 26 and x = 22
- 17. Book problems Solving Equations with Words activity. Update your flipbooks! Most of you have not done this lately. You are getting a grade for this. Worksheet