# Basic Algebra/Working with Numbers/Absolute Value

## Vocabulary

Absolute Value
The absolute value of a number is its distance from zero (0) on a number line. This action ignores the "+" or "–" sign of a number because distance in mathematics is never negative. The symbol ${\displaystyle |x|}$ represents the absolute value of ${\displaystyle x}$. It is also called modulus ${\displaystyle x}$.

## Lesson

The absolute value of a number is its distance from zero (0) on a number line. This action ignores the “+” or “–“ sign of a number because distance in mathematics is never negative.

You identify an absolute value of a number by writing the number between two vertical bars referred to as absolute value brackets: |number|.

A helpful way of thinking about absolute value is relating it to a railroad track. If you were to stand on a railroad track, more specifically on any one of the railroad ties and mark that spot as zero, railroad ties to the left would represent negative numbers and railroad ties to the right would represent positive numbers.

The number –7 is 7 units away from zero on the negative side of the railroad track. So, the following is true, |-7| = 7. The number 16 is 16 units away from zero on the positive side of the railroad track. So, |16| = 16. The number 0 is 0 units from zero on the railroad track. So |0| = 0 Therefore, the absolute value of any number is a positive number or zero.

In summary… THE ABSOLUTE VALUE OF A NUMBER

If x is a positive number, then ${\displaystyle |x|=x}$. Example: ${\displaystyle |5|=5}$

If x is zero, then ${\displaystyle |x|=0}$. Example: ${\displaystyle |0|=0}$

If x is a negative number, ${\displaystyle |x|=-x}$. Example: ${\displaystyle |-6|=-(-6)=6}$

You can find the absolute value of expressions as well. When addressed with this you must treat the absolute value brackets as you would parentheses. You need to simplify everything inside the absolute value brackets by performing all the necessary operations by following the order of operations. Your last step once you have a single number inside the absolute value brackets is to take the absolute value. For example, ${\displaystyle |-5+1\times 3|=|-5+3|=|-2|=2}$.

You may use the absolute value to find the distance between two numbers on the number line. Let a and b be variables. Then ${\displaystyle |a-b|}$ is the distance between a and b. For example, if ${\displaystyle a=3}$ and ${\displaystyle b=7}$, then ${\displaystyle |3-7|=|-4|=4}$. Because you used the absolute value, the distance is the same if you switch the order of the two numbers; if ${\displaystyle a=7}$ and ${\displaystyle b=3}$, then ${\displaystyle |7-3|=|4|=4}$.

Two things to watch out for are an opposite sign and/or an operation outside the absolute value brackets. As stated above, simplify everything inside the absolute value brackets by performing all the necessary operations by following the order of operations. Your last step once you have a single number inside the absolute value brackets is to take the absolute value. Once you have taken the absolute value then perform the other necessary operations by following the order of operations from left to right in the expression. For example, ${\displaystyle -|5|=-5}$ and ${\displaystyle 7+|-5+1\times 3|=7+|-5+3|=7+|-2|=7+2=9}$.

## Example Problems

• ${\displaystyle |0|=0}$
• ${\displaystyle -|-21|=-21}$
• ${\displaystyle |7-1|+9=|6|+9=6+9=15}$
• ${\displaystyle |0.5|=0.5}$
• ${\displaystyle |-5\times 3|=|-15|=15}$
• ${\displaystyle \left|{\frac {-2}{3}}\right|={\frac {2}{3}}}$
• ${\displaystyle |25-16|=|9|=9}$
• ${\displaystyle |9|=9}$
• ${\displaystyle |3.5-5.7|=|-2.2|=2.2}$
• ${\displaystyle |-11.5|=11.5}$
• ${\displaystyle 5-|3|=5-3=2}$
• ${\displaystyle -|6.7|=-6.7}$
• ${\displaystyle 8+|-6\times 2|=8+|-12|=8+12=20}$

## Practice Problems

Solve.

1

 |-3|=

2

 |6|=

3

 |-1.8|=

4

 |-27|=

5

 |5/7|=

6

 -|12|=

7

 |0|=

8

 |-3 + 8|=

9

 |9 + 3|=

10

 |-5 + 1|=

11

 |2 - 5|=

12

 |7| + 2=

13

 |13| - 21=

14

 |-5| - 1=

15

 3 - |-2|=

16

 9 + |-3|=

17

 |1 - 6| + 5=

18

 -5 - 1 + |6|=

19

 -6 + |-5 - 1|=

20

 12 - | 3(-5) + 6|=

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