- Simplify expressions with absolute value.
absolute value
Absolute value represents the distance of a number from zero on the number line, always resulting in a non-negative value. The absolute value of a number [latex]x[/latex] is written as [latex]|x|[/latex].
In simple terms: [latex]|x|[/latex] equals [latex]x[/latex] if [latex]x[/latex] is positive or zero, and equals [latex]-x[/latex] if [latex]x[/latex] is negative.
To simplify expressions with absolute value:
- Treat absolute value bars as parenthesis and simplify what’s inside the absolute value bars first
- Determine if the expression inside is positive, negative, or zero
- Apply the absolute value definition:
- If positive or zero: remove the bars, keep the number
- If negative: remove the bars, change the sign of the number
- Simplify any remaining arithmetic following the order of operations
Simplify each expression:
a) [latex]|7|[/latex]
Since [latex]7 > 0[/latex], we have [latex]|7| = 7[/latex]
b) [latex]|-12|[/latex]
Since [latex]-12 < 0[/latex], we have [latex]|-12| = 12[/latex]
c) [latex]|3 - 8|[/latex]
[latex]\begin{align} |3 - 8| &= |-5| \quad \text{(simplify inside first)} \\ &= 5 \quad \text{(since -5 < 0)} \end{align}[/latex]
d) [latex]2|4 - 9| + 3[/latex]
[latex]\begin{align} 2|4 - 9| + 3 &= 2|-5| + 3 \quad \text{(simplify inside first)} \\ &= 2(5) + 3 \quad \text{(since -5 < 0)} \\ &= 10 + 3 \\ &= 13 \end{align}[/latex]
When you see nested operations like [latex]|a - b|[/latex], always compute what’s inside the bars first, then apply the absolute value. Don’t try to “distribute” the absolute value bars to individual terms.