Module 1: Algebraic Expressions, Linear Equations and Mathematical Models
1.2 – Absolute Value
Learning Objectives
(1.2.1) – Evaluating expressions with absolute value signs
(1.2.1) – Evaluating expressions with absolute value signs
Absolute value: a number’s distance from zero; it’s always positive:
|3|=3
|−5|=5
|0|=0
Recall that the absolute value of a quantity is always positive or 0.
Example
Find |7−10|.
Show Answer
|7−10|=|−3|=3
Try It
Example
Find −|3−2|.
Show Answer
−|3−2|=−|−1|=−1
Try It
When you see an absolute value expression included within a larger expression, treat the absolute value like a grouping symbol and evaluate the expression within the absolute value sign first. Then take the absolute value of that expression. The example below shows how this is done.
Example
Simplify 3+|2−6|2|3⋅1.5|−(−3).
Show Solution
This problem has absolute values, decimals, multiplication, subtraction, and addition in it.
Grouping symbols, including absolute value, are handled first. Simplify the numerator, then the denominator.
Evaluate |2–6|.
3+|2−6|2|3⋅1.5|−(−3)3+|−4|2|3⋅1.5|−(−3)
Take the absolute value of |−4|.
3+|−4|2|3⋅1.5|−(−3)3+42|3⋅1.5|−(−3)
Add the numbers in the numerator.
3+42|3⋅1.5|−(−3)72|3⋅1.5|−(−3)
Now that the numerator is simplified, turn to the denominator.
Evaluate the absolute value expression first. 3⋅1.5=4.5, giving
72|3⋅1.5|−(−3)72|4.5|−(−3)
The expression “2|4.5|” reads “2 times the absolute value of 4.5.” Multiply 2 times 4.5.
72|4.5|−(−3)79−(−3)
Subtract.
79−(−3)712
Answer
3+|2−6|2|3⋅1.5|−3(−3)=712
The following video uses the order of operations to simplify an expression in fraction form that contains absolute value terms. Note how the absolute values are treated like parentheses and brackets when using the order of operations.