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

Key Concepts

Absolute Value

Absolute value is a mathematical function that measures the distance of a number from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|. For any real number x, |x| is equal to x if x is positive or zero, and -x if x is negative. Understanding absolute value is crucial for solving equations that involve it, as it leads to two possible cases.

Solving Absolute Value Equations

To solve an absolute value equation like |x| = 8, one must recognize that the equation can be split into two separate cases: x = 8 and x = -8. This is because the absolute value of a number can be equal to a positive value in two different scenarios. Solving these cases will yield the complete set of solutions for the equation.

No Solution Condition

In some cases, an absolute value equation may have no solution. This occurs when the absolute value is set equal to a negative number, as absolute values cannot be negative. For example, the equation |x| = -5 has no solution because there is no real number whose absolute value can equal a negative value. Recognizing when an equation has no solution is essential for accurately solving absolute value problems.