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.

Verified step by step guidance

1. insert step 1> Start by understanding the properties of absolute value. The absolute value of a number, |x|, is always non-negative, meaning it is either positive or zero.
2. insert step 2> Analyze the given equation |x| = -6. Since the absolute value cannot be negative, this equation has no solution.
3. insert step 3> Consider the statement x = 6 or x = -6. This implies that the absolute value of x is 6, which would be written as |x| = 6, not -6.
4. insert step 4> Conclude that the original statement is false because |x| = -6 has no solution, and the correct equivalent statement should be |x| = 6, which leads to x = 6 or x = -6.
5. insert step 5> Therefore, the necessary change to make the statement true is to replace -6 with 6 in the equation, resulting in |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.