Учебные пособия > Prealgebra

This figure is a number line. The points negative 5 and 5 are labeled. <a href=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." />

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.

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[/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

  1. [latex]|x|\text< when >x=-35[/latex]
  2. [latex]|\mathit>|\text< when >y=-20[/latex]
  3. [latex]-|u|\text< when >u=12[/latex]
  4. [latex]-|p|\text< when >p=-14[/latex]
1. To find [latex]|x|[/latex] when [latex]x=-35:[/latex]
[latex]|x|[/latex]
Substitute [latex]\color[/latex] for x. [latex]\mid\color\mid[/latex]
Take the absolute value. [latex]35[/latex]
inside of parentheses. The third row states to simplify and has 20 inside of the absolute value symbol. The fourth row states to take the absolute value and has 20.">
2. To find [latex]|-y|[/latex] when [latex]y=-20:[/latex]
[latex]|-y|[/latex]
Substitute [latex]\color[/latex] for y. [latex]\mid-(\color)\mid[/latex]
Simplify. [latex]|20|[/latex]
Take the absolute value. [latex]20[/latex]
12 for u and has a negative sign in front of the absolute value symbol with 12 on the inside. The third row states to take the absolute value and has negative 12.">
3. To find [latex]-|u|[/latex] when [latex]u=12:[/latex]
[latex]-|u|[/latex]
Substitute [latex]\color[/latex] for u. [latex]-\mid\color\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[/latex] for p. [latex]-\mid\color\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

  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

  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

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