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Mastering Absolute Value Equations: A Comprehensive Guide

Learn how to solve absolute value equations effectively using mental math techniques. This guide breaks down the process, illustrating how the expression within absolute value symbols can be both positive and negative. Through step-by-step examples, such as solving |x - 2| = 5, we demonstrate isolating the absolute value expression, leading to multiple solutions. Understand the properties of absolute values and how they reflect distances from zero on the number line to enhance your problem-solving skills in mathematics.

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Mastering Absolute Value Equations: A Comprehensive Guide

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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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