# 1 7 Solving AbsoluteValue Equations Objectives Solve equations

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1 -7 Solving Absolute-Value Equations Objectives Solve equations in one variable that contain absolute-value expressions. Holt Mc. Dougal Algebra 1

1 -7 Solving Absolute-Value Equations Holt Mc. Dougal Algebra 1

1 -7 Solving Absolute-Value Equations Holt Mc. Dougal Algebra 1

1 -7 Solving Absolute-Value Equations To solve absolute-value equations, perform inverse operations to isolate the absolute-value expression on one side of the equation. Then you must consider two cases. Holt Mc. Dougal Algebra 1

1 -7 Solving Absolute-Value Equations Additional Example 1 A: Solving Absolute-Value Equations Solve the equation. |x| = 12 Think: What numbers are 12 units from 0? • 12 units 12 10 8 6 4 2 Case 1 x = 12 Case 2 x = – 12 • 0 12 units 2 6 • 8 10 12 Rewrite the equation as two cases. The solutions are {12, – 12}. Holt Mc. Dougal Algebra 1 4

1 -7 Solving Absolute-Value Equations Additional Example 1 B: Solving Absolute-Value Equations Solve the equation. 3|x + 7| = 24 |x + 7| = 8 Case 2 Case 1 x + 7 = 8 x + 7 = – 8 – 7 – 7 x =1 x = – 15 The solutions are {1, – 15}. Holt Mc. Dougal Algebra 1

1 -7 Solving Absolute-Value Equations Check It Out! Example 1 a Solve the equation. |x| – 3 +3 |x| Case 1 x=7 =4 =4 +3 =7 Case 2 x = – 7 The solutions are {7, – 7}. Holt Mc. Dougal Algebra 1

1 -7 Solving Absolute-Value Equations Check It Out! Example 1 b Solve the equation. 8 =|x 2. 5| Case 1 8 = x 2. 5 +2. 5 10. 5 = x Case 2 8 = x 2. 5 +2. 5 5. 5 = x The solutions are {10. 5, – 5. 5}. Holt Mc. Dougal Algebra 1

1 -7 Solving Absolute-Value Equations The table summarizes the steps for solving absolute-value equations. Solving an Absolute-Value Equation 1. Use inverse operations to isolate the absolute-value expression. 2. Rewrite the resulting equation as two cases that do not involve absolute values. 3. Solve the equation in each of the two cases. Holt Mc. Dougal Algebra 1

1 -7 Solving Absolute-Value Equations Not all absolute-value equations have two solutions. If the absolute-value expression equals 0, there is one solution. If an equation states that an absolute-value is negative, there are no solutions. Holt Mc. Dougal Algebra 1

1 -7 Solving Absolute-Value Equations Additional Example 2 A: Special Cases of Absolute. Value Equations Solve the equation. 8 = |x + 2| 8 +8 +8 0 = |x + 2| 0= x+2 2 = x The solution is { 2}. Holt Mc. Dougal Algebra 1

1 -7 Solving Absolute-Value Equations Additional Example 2 B: Special Cases of Absolute. Value Equations Solve the equation. 3 + |x + 4| = 0 3 3 |x + 4| = 3 This equation has no solution. Holt Mc. Dougal Algebra 1

1 -7 Solving Absolute-Value Equations Remember! Absolute value must be nonnegative because it represents a distance. Holt Mc. Dougal Algebra 1

1 -7 Solving Absolute-Value Equations Check It Out! Example 2 a Solve the equation. 2 |2 x 5| = 7 2 2 |2 x 5| = 5 This equation has no solution. Holt Mc. Dougal Algebra 1

1 -7 Solving Absolute-Value Equations Check It Out! Example 2 b Solve the equation. 6 + |x 4| = 6 +6 +6 |x 4| = 0 x 4 = 0 + 4 +4 x Holt Mc. Dougal Algebra 1 = 4

1 -7 Solving Absolute-Value Equations Lesson Quiz Solve each equation. 1. 15 = |x| – 15, 15 2. 2|x – 7| = 14 0, 14 3. |x + 1|– 9 = – 9 – 1 4. |5 + x| – 3 = – 2 – 6, – 4 5. 7 + |x – 8| = 6 no solution 6. Inline skates typically have wheels with a diameter of 74 mm. The wheels are manufactured so that the diameters vary from this value by at most 0. 1 mm. Write and solve an absolute-value equation to find the minimum and maximum diameters of the wheels. |x – 74| = 0. 1; 73. 9 mm; 74. 1 mm Holt Mc. Dougal Algebra 1