In Exercises 137–140, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation |x| = - 6 is equivalent to x = 6 or x = - 6.

Key Concepts

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|. For any real number x, the absolute value is always non-negative, meaning |x| ≥ 0. Therefore, an equation like |x| = -6 has no solution, as absolute values cannot equal negative numbers.

Equivalence of Equations

Two equations are equivalent if they have the same solution set. In this case, the statement claims that |x| = -6 is equivalent to x = 6 or x = -6. However, since |x| cannot be negative, the original equation has no solutions, making the equivalence false. Understanding equivalence is crucial for validating statements in algebra.

True and False Statements in Algebra

In algebra, determining the truth value of a statement involves verifying if the statement holds under the defined conditions. A false statement can often be corrected by altering its components. In this case, the statement is false because it incorrectly asserts that |x| can equal a negative number, which is impossible. Recognizing true and false statements is essential for logical reasoning in mathematics.