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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.