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.

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Recall the definition of absolute value: for any real number \(x\), \(|x|\) represents the distance of \(x\) from zero on the number line, and distance is always non-negative. Therefore, \(|x| \geq 0\) for all real \(x\).

Analyze the given equation \(|x| = -6\). Since the absolute value cannot be negative, this equation has no real solutions.

The statement claims that \(|x| = -6\) is equivalent to \(x = 6\) or \(x = -6\). This is false because the right side implies solutions exist, but the left side has no solutions.

To make the statement true, change the equation to \(|x| = 6\), which is a valid absolute value equation with solutions \(x = 6\) or \(x = -6\).

Thus, the corrected true statement is: The equation \(|x| = 6\) is equivalent to \(x = 6\) or \(x = -6\).

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Absolute Value Definition

The absolute value of a number is its distance from zero on the number line, always non-negative. For any real number x, |x| ≥ 0, meaning absolute value cannot be negative. This is crucial to understand why |x| = -6 has no solution.

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Solving Absolute Value Equations

When solving equations like |x| = a, where a ≥ 0, the solutions are x = a or x = -a. If a is negative, the equation has no solution because absolute value cannot equal a negative number.

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Logical Equivalence and Statement Correction

Determining if a statement is true or false involves checking its logical equivalence. If false, identify the error and correct it. Here, the statement |x| = -6 is false; the corrected true statement would be that |x| = 6 is equivalent to x = 6 or x = -6.

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