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S OLVING A BSOLUTE- V ALUE E QUATIONS

S OLVING A BSOLUTE- V ALUE E QUATIONS. You can solve some absolute-value equations using mental math. For instance, you learned that the equation | x |  8 has two solutions: 8 and  8.

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S OLVING A BSOLUTE- V ALUE E QUATIONS

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  1. SOLVING ABSOLUTE-VALUE EQUATIONS You can solve some absolute-value equations using mental math. For instance, you learned that the equation |x| 8 has two solutions: 8 and 8. To solve absolute-value equations, you can use the fact that the expression inside the absolute value symbols can be either positive or negative.

  2. Solving an Absolute-Value Equation The expressionx  2 can be equal to 5 or 5. x  2IS NEGATIVE | x  2 |  5 x  2IS POSITIVE x  2 IS POSITIVE x  2 IS POSITIVE x  2 IS NEGATIVE x  2  5 | x  2 |  5 | x  2 |  5 x  3 x  2  5 x  2  5 x  2  5 x  2  5 CHECK x  7 x  7 Solve | x  2 |  5 Solve | x  2 |  5 SOLUTION The expressionx  2can be equal to5or5. x  2 IS POSITIVE x  2 IS NEGATIVE | x  2 |  5 | x  2 |  5 x  2  5 x  2  5 x  7 x  3 The equation has two solutions: 7 and –3. | 7  2 |  | 5 |  5 |3  2 |  | 5 |  5

  3. Solving an Absolute-Value Equation Isolate theabsolute value expressionon one side of the equation. 2x  7 IS NEGATIVE 2x  7 IS POSITIVE 2x  7 IS NEGATIVE 2x  7 IS POSITIVE 2x  7 IS POSITIVE | 2x  7 |  5  4 | 2x  7 |  5  4 | 2x  7 |  5  4 | 2x  7 |  9 | 2x  7 |  9 | 2x  7 |  9 2x  7  9 2x  7  +9 2x  7  9 2x  7  +9 2x  7  +9 2x  2 2x  16 2x  16 x  8 x  1 x  1 TWO SOLUTIONS x  8 x  8 Solve | 2x  7 |  5  4 Solve | 2x  7 |  5  4 SOLUTION Isolate theabsolute value expressionon one side of the equation. 2x  7 IS POSITIVE 2x  7 IS NEGATIVE | 2x  7 |  5  4 | 2x  7 |  5  4 | 2x  7 |  9 | 2x  7 |  9 2x  7  +9 2x  7  9 2x  16 2x  2 x  8 x  1

  4. Solving an Absolute-Value Equation Recall that |x |is the distance between x and 0. If |x |  8, then any number between 8 and 8 is a solution of the inequality.  8  7  6  5  4  3  2  1 0 1 2 3 4 5 6 7 8 Recall that xis the distance between x and 0. If x 8, then any number between 8 and 8 is a solution of the inequality. Recall that |x |is the distance between x and 0. If |x |  8, then any number between 8 and 8 is a solution of the inequality. You can use the following properties to solve absolute-value inequalities and equations.

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