In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. |x| = 8

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Recognize that the equation involves an absolute value expression: $|x| = 8$.

Recall the definition of absolute value: $|x|$ represents the distance of $x$ from zero on the number line, which is always non-negative.

Set up two separate equations to solve for $x$ because $|x| = a$ implies $x = a$ or $x = -a$ when $a \geq 0$.

Write the two equations: $x = 8$ and $x = -8$.

Solve each equation individually to find the solutions for $x$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Absolute Value Definition

The absolute value of a number represents its distance from zero on the number line, always as a non-negative value. For any real number x, |x| equals x if x is positive or zero, and -x if x is negative.

Solving Absolute Value Equations

To solve an equation like |x| = a (where a ≥ 0), set up two separate equations: x = a and x = -a. This accounts for both positive and negative values whose absolute value equals a.

No Solution Condition

If the absolute value equation is set equal to a negative number (e.g., |x| = -5), there is no solution because absolute values cannot be negative. Recognizing this prevents incorrect or impossible solutions.

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