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

Verified step by step guidance

1. Recognize that the equation involves an absolute value expression: \(|x| = 8\).
2. Recall the definition of absolute value: \(|x|\) represents the distance of \(x\) from zero on the number line, which is always non-negative.
3. Set up two separate equations to solve for \(x\) because \(|x| = a\) implies \(x = a\) or \(x = -a\) when \(a \geq 0\).
4. Write the two equations: \(x = 8\) and \(x = -8\).
5. Solve each equation individually to find the solutions for \(x\).

Key Concepts

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.