Expressions Involving Absolute Value
Absolute value is a number's distance from zero on the number line. Distance is never negative, so absolute value is never negative: and , because both numbers sit nine steps from zero.
When absolute value bars appear inside a bigger expression, treat them like parentheses — finish everything inside the bars first, take the absolute value, and then continue with the rest of the order of operations. That one habit handles every problem in this lesson.
The bars act like parentheses
Absolute value bars are grouping symbols. In , the subtraction inside the bars happens first, then the absolute value, then the multiplication by , then the addition of .
One warning: the bars only protect what is inside them. In , the inner becomes , but the negative sign outside the bars is untouched, so the value is . Absolute value makes the inside non-negative — it does nothing to signs on the outside.
Evaluating with substitution
When the expression has a variable, substitute the value first, then follow the order above. To evaluate at : inside the bars, . The absolute value gives . Then .
Substitute negative values inside parentheses — writing instead of costs nothing and prevents the sign slips that come from squeezing a negative next to a minus sign.
Worked examples
Example 1: two absolute values
Evaluate .
Answer:
Example 2: substitute, then evaluate
Evaluate when .
Answer:
Example 3: two variables inside the bars
Evaluate when and .
Answer:
Try one yourself
Common questions
Can an absolute value ever be negative?
The absolute value itself, never — it is a distance. But an expression containing absolute value can be negative. . The bars only control what is between them.
Is equal to or ?
. Work from the inside out: , and the negative sign outside the bars still applies, giving . The bars do not reach signs that sit outside them.
Why do I evaluate inside the bars before taking the absolute value?
Because the bars are a grouping symbol, like parentheses. asks for the distance of the result from zero, so you need the result first: , then . Taking absolute values term by term — — is a different (wrong) calculation.
What is ?
. Zero sits zero steps from itself. It is the one number whose absolute value is not positive — which is why the careful phrasing is that absolute value is never negative, rather than always positive.
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