Determine whether each statement is true or false. |25| = |-25|

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. Evaluate the absolute value of 25: \(|25|\) is the distance of 25 from zero, which is simply 25.
3. Evaluate the absolute value of -25: \(|-25|\) is the distance of -25 from zero, which is also 25 because distance is always positive.
4. Compare the two values: since both \(|25|\) and \(|-25|\) equal 25, they are equal.
5. Conclude that the statement \(|25| = |-25|\) is true because absolute value ignores the sign of the number.

Key Concepts

Absolute Value Definition

The absolute value of a number is its distance from zero on the number line, always expressed as a non-negative value. For any real number x, |x| equals x if x is positive or zero, and -x if x is negative.

Properties of Absolute Value

Absolute value has key properties such as |a| ≥ 0 for any real number a, and |a| = |-a|, meaning the absolute value of a number is equal to the absolute value of its opposite. This property is essential for comparing expressions involving absolute values.

Evaluating Absolute Value Expressions

To evaluate an absolute value expression, substitute the number inside the bars and apply the definition. For example, |25| = 25 because 25 is positive, and |-25| = 25 because the negative sign is negated by the absolute value.