Determine whether each statement is true or false. |5+(-13) | = |5| + |-13|

Verified step by step guidance

1. Recall the definition of absolute value: for any real number \(a\), \(|a|\) represents the distance of \(a\) from zero on the number line, and it is always non-negative.
2. Evaluate the left side of the equation: calculate \(|5 + (-13)|\). First, perform the addition inside the absolute value: \$5 + (-13) = 5 - 13$.
3. Simplify the sum inside the absolute value: \$5 - 13 = -8\(, so the left side becomes \)|-8|$.
4. Evaluate the right side of the equation: calculate \(|5| + |-13|\). Find the absolute values separately: \(|5|\) and \(|-13|\).
5. Compare the two sides: check if \(|-8|\) is equal to \(|5| + |-13|\). Since \(|-8| = 8\), \(|5| = 5\), and \(|-13| = 13\), determine if \$8 = 5 + 13$.

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 example, |5| = 5 and |-13| = 13, regardless of the sign of the original number.

Properties of Absolute Value

Absolute value has specific properties, such as |a| ≥ 0 and |a| = |-a|. Importantly, the absolute value of a sum is not generally equal to the sum of the absolute values, i.e., |a + b| ≠ |a| + |b| in most cases.

Evaluating Expressions with Absolute Values

To evaluate expressions involving absolute values, first simplify inside the absolute value, then apply the absolute value operation. For example, |5 + (-13)| = |-8| = 8, while |5| + |-13| = 5 + 13 = 18, showing the two sides can differ.