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

Verified step by step guidance

1. Understand the definition of absolute value: The absolute value of a number \( x \), denoted as \( |x| \), is the non-negative value of \( x \) without regard to its sign.
2. Recognize the property of square roots: The square root of \( x^2 \), denoted as \( \sqrt{x^2} \), is also the non-negative value of \( x \).
3. Compare the expressions: Both \( |x| \) and \( \sqrt{x^2} \) yield the same non-negative value for any real number \( x \).
4. Conclude that for any real number \( x \), \( |x| = \sqrt{x^2} \) holds true, meaning the equation is satisfied for all real numbers.
5. Thus, the equation \( |x| = \sqrt{x^2} \) has infinitely many solutions because it is true for every real number \( x \).

Key Concepts

Absolute Value

The absolute value of a number, denoted as |x|, represents its distance from zero on the number line, regardless of direction. This means |x| is always non-negative. For example, |3| = 3 and |-3| = 3. Understanding absolute value is crucial for solving equations involving it, as it leads to two possible cases based on the sign of x.

Square Root and Squaring

The square root of a number, represented as √x², is the value that, when multiplied by itself, gives the original number. Since squaring any real number (positive or negative) results in a non-negative value, √x² equals |x|. This relationship is fundamental in understanding how the equation |x| = √x² holds true for all real numbers.

Infinitely Many Solutions

An equation has infinitely many solutions when there are countless values that satisfy it. In the case of |x| = √x², since both sides are equal for all real numbers x, every real number is a solution. This concept is essential in algebra, as it indicates that the solution set is not limited to discrete values but encompasses a continuous range.