For what value(s) of x is |x| = 4 true?

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Understand the meaning of the absolute value function: \(|x|\) represents the distance of \(x\) from zero on the number line, and it is always non-negative.

Set up the equation given: \(|x| = 4\) means we are looking for all values of \(x\) whose distance from zero is 4.

Recall the definition of absolute value: \(|x| = a\) implies \(x = a\) or \(x = -a\) when \(a \geq 0\).

Apply this to our problem: since \(a = 4\), the solutions are \(x = 4\) or \(x = -4\).

Conclude that the values of \(x\) satisfying \(|x| = 4\) are \(x = 4\) and \(x = -4\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Absolute Value Definition

The absolute value of a number represents its distance from zero on the number line, regardless of direction. For any real number x, |x| is always non-negative. For example, |4| = 4 and |-4| = 4.

Solving Absolute Value Equations

An equation involving absolute value, such as |x| = a, where a ≥ 0, has two solutions: x = a and x = -a. This is because both values have the same distance from zero, satisfying the absolute value condition.

Properties of Real Numbers

Understanding that real numbers include both positive and negative values is essential. When solving |x| = 4, recognizing that x can be either positive 4 or negative 4 relies on this property of real numbers.

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