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

Verified step by step guidance

1. 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.
2. 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.
3. 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.
4. Because every real number satisfies this equation, there are infinitely many solutions, as the set of real numbers is infinite.
5. Thus, the equation \(|x| = \sqrt{x^2}\) is true for all \(x \in \mathbb{R}\), explaining why it has infinitely many solutions.

Key Concepts

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.