Explain why the equation | x | = √x² has infinitely many solutions.

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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, which is always non-negative.

Understand that the square root of \(x^2\) is defined as \(\sqrt{x^2} = |x|\) because squaring \(x\) makes it non-negative, and the square root returns the principal (non-negative) root.

Since \(|x|\) and \(\sqrt{x^2}\) are equivalent expressions for all real numbers \(x\), the equation \(|x| = \sqrt{x^2}\) holds true for every real number.

Because every real number satisfies this equation, there are infinitely many solutions, as the set of real numbers is infinite.

Thus, the equation \(|x| = \sqrt{x^2}\) is true for all \(x \in \mathbb{R}\), explaining why it has infinitely many solutions.

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주요 개념

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Absolute Value Definition

The absolute value of a number x, denoted |x|, is the non-negative value of x without regard to its sign. It is defined as |x| = x if x ≥ 0, and |x| = -x if x < 0. This means |x| is always zero or positive.

Square Root and Squaring Relationship

The square root of x², written as √(x²), equals the non-negative value of x because squaring any real number makes it non-negative. Thus, √(x²) = |x|, which shows the two expressions are equivalent for all real x.

Infinite Solutions in Real Numbers

Since |x| = √(x²) holds true for every real number x, the equation has infinitely many solutions. This is because both sides represent the same value for all x in the real number set, not just specific points.