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Study Guides > Prealgebra

Simplifying Expressions With Absolute Value

Learning Outcomes

  • Use correct notation to indicate absolute value
  • Simplify expressions that contain absolute value
  • Evaluate expressions that contain absolute value
  We saw that numbers such as [latex]5[/latex] and [latex]-5[/latex] are opposites because they are the same distance from [latex]0[/latex] on the number line. They are both five units from [latex]0[/latex]. The distance between [latex]0[/latex] and any number on the number line is called the absolute value of that number. Because distance is never negative, the absolute value of any number is never negative. The symbol for absolute value is two vertical lines on either side of a number. So the absolute value of [latex]5[/latex] is written as [latex]|5|[/latex], and the absolute value of [latex]-5[/latex] is written as [latex]|-5|[/latex] as shown below. This figure is a number line. The points negative 5 and 5 are labeled. Above the number line the distance from negative 5 to 0 is labeled as 5 units. Also above the number line the distance from 0 to 5 is labeled as 5 units.

Absolute Value

The absolute value of a number is its distance from [latex]0[/latex] on the number line. The absolute value of a number [latex]n[/latex] is written as [latex]|n|[/latex]. [latex-display]|n|\ge 0\text{for all numbers}[/latex-display]
   

example

Simplify: [latex-display]|3|[/latex-display] [latex-display]|-44|[/latex-display] [latex-display]|0|[/latex-display] Solution:
1.
[latex]|3|[/latex]
[latex]3[/latex] is [latex]3[/latex] units from zero. [latex]3[/latex]
2.
[latex]|-44|[/latex]
[latex]−44[/latex] is [latex]44[/latex] units from zero. [latex]44[/latex]
3.
[latex]|0|[/latex]
[latex]0[/latex] is already at zero. [latex]0[/latex]
 
  In the video below we show more example of how to find the absolute value of an integer. https://youtu.be/I8bTqGmkqGI We treat absolute value bars just like we treat parentheses in the order of operations. We simplify the expression inside first.

example

Evaluate:
  1. [latex]|x|\text{ when }x=-35[/latex]
  2. [latex]|\mathit{\text{-y}}|\text{ when }y=-20[/latex]
  3. [latex]-|u|\text{ when }u=12[/latex]
  4. [latex]-|p|\text{ when }p=-14[/latex]

Answer: Solution:

1. To find [latex]|x|[/latex] when [latex]x=-35:[/latex]
[latex]|x|[/latex]
Substitute [latex]\color{red}{-35}[/latex] for x. [latex]\mid\color{red}{-35}\mid[/latex]
Take the absolute value. [latex]35[/latex]
2. To find [latex]|-y|[/latex] when [latex]y=-20:[/latex]
[latex]|-y|[/latex]
Substitute [latex]\color{red}{-20}[/latex] for y. [latex]\mid-(\color{red}{-20})\mid[/latex]
Simplify. [latex]|20|[/latex]
Take the absolute value. [latex]20[/latex]
3. To find [latex]-|u|[/latex] when [latex]u=12:[/latex]
[latex]-|u|[/latex]
Substitute [latex]\color{red}{12}[/latex] for u. [latex]-\mid\color{red}{12}\mid[/latex]
Take the absolute value. [latex]-12[/latex]
4. To find [latex]-|p|[/latex] when [latex]p=-14:[/latex]
[latex]-|p|[/latex]
Substitute [latex]\color{red}{-14}[/latex] for p. [latex]-\mid\color{red}{-14}\mid[/latex]
Take the absolute value. [latex]-14[/latex]

  Notice that the result is negative only when there is a negative sign outside the absolute value symbol.  

example

Fill in [latex]\text{<},\text{>},\text{or}=[/latex] for each of the following:
  1. [latex]|-5|___-|-5|[/latex]
  2. [latex]8___-|-8|[/latex]
  3. [latex]-9___-|-9|[/latex]
  4. [latex]-|-7|___ - 7[/latex]

Answer: Solution: To compare two expressions, simplify each one first. Then compare.

1.
[latex]|-5|___-|-5|[/latex]
Simplify. [latex]5___ - 5[/latex]
Order. [latex]5>-5[/latex]
2.
[latex]8___-|-8|[/latex]
Simplify. [latex]8___ - 8[/latex]
Order. [latex]8>-8[/latex]
3.
[latex]-9___-|-9|[/latex]
Simplify. [latex]-9___ - 9[/latex]
Order. [latex]-9=-9[/latex]
4.
[latex]-|-7|___ - 7[/latex]
Simplify. [latex]-7___ - 7[/latex]
Order. [latex]-7=-7[/latex]

  In the video below we show more examples of how to compare expressions that include absolute value and integers. https://youtu.be/TendEcSaM3w Absolute value bars act like grouping symbols. First simplify inside the absolute value bars as much as possible. Then take the absolute value of the resulting number, and continue with any operations outside the absolute value symbols.

example

Simplify:
  1. [latex]|9 - 3|[/latex]
  2. [latex]4|-2|[/latex]

Answer: Solution: For each expression, follow the order of operations. Begin inside the absolute value symbols just as with parentheses.

1.
[latex]|9−3|[/latex]
Simplify inside the absolute value sign. [latex]|6|[/latex]
Take the absolute value. [latex]6[/latex]
2.
[latex]4|−2|[/latex]
Take the absolute value. [latex]4⋅2[/latex]
Multiply. [latex]8[/latex]

   

example

Simplify: [latex]|8+7|-|5+6|[/latex].

Answer: Solution: For each expression, follow the order of operations. Begin inside the absolute value symbols just as with parentheses.

[latex]|8+7|−|5+6|[/latex]
Simplify inside each absolute value sign. [latex]|15|−|11|[/latex]
Subtract. [latex]4[/latex]

   

example

Simplify: [latex]24-|19 - 3\left(6 - 2\right)|[/latex].

Answer: Solution: We use the order of operations. Remember to simplify grouping symbols first, so parentheses inside absolute value symbols would be first.

[latex]24-|19 - 3\left(6 - 2\right)|[/latex]
Simplify in the parentheses first. [latex]24-|19 - 3\left(4\right)|[/latex]
Multiply [latex]3\left(4\right)[/latex] . [latex]24-|19 - 12|[/latex]
Subtract inside the absolute value sign. [latex]24-|7|[/latex]
Take the absolute value. [latex]24 - 7[/latex]
Subtract. [latex]17[/latex]

 

try it

  [ohm_question]145010[/ohm_question]  
Watch the following video to see more examples of how to simplify expressions that contain absolute value. https://youtu.be/cd24nT7mAi0

Licenses & Attributions

CC licensed content, Original

  • Simplify Absolute Value Expressions. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.

CC licensed content, Shared previously

  • Ex 1: Determine the Absolute Value of an Integer. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
  • Ex: Comparing Absolute Value of Integers. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
  • Question ID: 144930, 144931, 144932, 144933, 144934, 144979, 145007, 145009. Authored by: Alyson Day. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.

CC licensed content, Specific attribution