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. S OLVING.

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Solving an Absolute Value Equation

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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. S OLVING A BSOLUTE- V ALUE E QUATIONS To solve absolute-value equations, you can use the fact that the expression inside the absolute value symbols can be either positive or negative. So, to solve an absolute value equation: 1.isolate the absolute value part of the equation and 2.set the absolute value expression equal to both the positive and negative solutions and solve, then 3.check your answers.

Solving an Absolute-Value Equation Solve | x  2 |  5 x  2 IS POSITIVE | x  2 |  5 x  7 x  3x  3 x  2 IS NEGATIVE | x  2 |  5 | 7  2 |  | 5 |  5|  3  2 |  |  5 |  5 The expression x  2 can be equal to 5 or  5. x  2   5 x  2 IS POSITIVE x  2   5 Solve | x  2 |  5 The expression x  2 can be equal to 5 or  5. S OLUTION x  2   5 x  2 IS POSITIVE | x  2 |  5 x  2   5 x  7 x  2 IS POSITIVE | x  2 |  5 x  2   5 x  7 x  2 IS NEGATIVE x  2   5 x  3x  3 x  2 IS NEGATIVE | x  2 |  5 x  2   5 The equation has two solutions: 7 and –3. C HECK

Solving an Absolute-Value Equation Solve | 2x  7 |  5  4 2x  7 IS POSITIVE | 2x  7 |  5  4 | 2x  7 |  9 2x  7  +9 2x  16 2x  7 IS NEGATIVE | 2x  7 |  5  4 | 2x  7 |  9 2x  7   9 2x  22x  2 x  1x  1 Isolate the absolute value expression on one side of the equation. S OLUTION Solve | 2x  7 |  5  4 2x  7 IS POSITIVE 2x  7  +9 2x  7 IS NEGATIVE 2x  7   9 2x  7 IS POSITIVE | 2x  7 |  5  4 | 2x  7 |  9 2x  7  +9 2x  16 x  8 2x  7 IS POSITIVE | 2x  7 |  5  4 | 2x  7 |  9 2x  7  +9 2x  16 x  8 2x  7 IS NEGATIVE | 2x  7 |  5  4 | 2x  7 |  9 2x  7   9 2x  22x  2 x  1x  1 TWO SOLUTIONS x  8 x  1x  1

Solving an Absolute-Value Equation The expression 9 - 2x can be equal to x or -(10 + 3x) S OLUTION Solve | 9 – 2x |  x 9 - 2x IS POSITIVE | 9 – 2x |  x 9 – 2x = x 9 = x -1 = 5x 9 - 2x IS NEGATIVE | 9 – 2x | = -(10 + 3x) 9 – 2x = -10 – 3x 9 + x = -10 TWO possible SOLUTIONS -19 is EXTRANEOUS because it doesn’t work when you check it!