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2.12: Absolute Value

Difficulty Level: Advanced Created by: CK-12
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What is the absolute value represented by the number line below?

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Khan Academy Absolute Value Equations

Guidance

Absolute value in the real number system is the distance from zero on the number line. It is always a positive number.

Absolute value is always written as \begin{align*}|x|\end{align*}|x|. Using this notation can be translated as “the positive value of \begin{align*}x\end{align*}x”.

Example A

\begin{align*}|-7|=?\end{align*}|7|=?

Solution:

\begin{align*}|-7|=7\end{align*}|7|=7

Example B

\begin{align*}|7-15|=?\end{align*}|715|=?

Solution:

\begin{align*}|7-15|&=?\\ |-8|&=?\end{align*}|715||8|=?=?

\begin{align*}|-8|=8\end{align*}|8|=8

Example C

\begin{align*}|-9|-|-2| = ?\end{align*}|9||2|=?

Solution:

\begin{align*}|-9|-|-2| &= ?\\ |-9|-|-2| &= 9-2\\ |-9|-|-2| &= 7\end{align*}|9||2||9||2||9||2|=?=92=7

Concept Problem Revisited

What is the absolute value represented by the number line below?

Remember the definition of absolute value in the real number system is the distance from zero on the number line. Let’s look at the distance from the -6 to zero.

Therefore \begin{align*}|-6|=6\end{align*}|6|=6.

Vocabulary

Absolute Value
Absolute value in the real number system is the distance from zero on the number line. It is always a positive number and is represented using the symbol \begin{align*}|x|\end{align*}|x|.

Guided Practice

1. \begin{align*}|-50|=?\end{align*}|50|=?

2. \begin{align*}\big |- \frac{4}{5} \big |=?\end{align*}45=?

3. \begin{align*}|-3|-|-4|=?\end{align*}|3||4|=?

Answers:

1. \begin{align*}|-50|=?\end{align*}|50|=?

\begin{align*}|-50|=50\end{align*}|50|=50

2. \begin{align*}\big | -\frac{4}{5} \big |=?\end{align*}45=?

\begin{align*}\big | - \frac{4}{5}\big |=\frac{4}{5}\end{align*}45=45

3.

\begin{align*}|-3|-|-4| &=?\\ |-3|-|-4| &=3-4\\ |-3|-|-4| &= -1\end{align*}|3||4||3||4||3||4|=?=34=1

Practice

Evaluate each of the following using a number line:

  1. \begin{align*}|-4.5|=?\end{align*}|4.5|=?
  2. \begin{align*}|-1|=?\end{align*}|1|=?
  3. \begin{align*}\big | -\frac{1}{3} \big |=?\end{align*}13=?
  4. \begin{align*}|4|=?\end{align*}|4|=?
  5. \begin{align*}|-2|=?\end{align*}|2|=?

Evaluate each of the following:

  1. \begin{align*}|-1|+|3|=?\end{align*}|1|+|3|=?
  2. \begin{align*}|5|-|-2|=?\end{align*}|5||2|=?
  3. \begin{align*}\big | -\frac{1}{2}\big |+ \big | - \frac{2}{3} \big |=?\end{align*}12+23=?
  4. \begin{align*}|-2.4|-|-1.6|=?\end{align*}|2.4||1.6|=?
  5. \begin{align*}|-3|-|-2.4|=?\end{align*}|3||2.4|=?

Evaluate each of the following:

  1. \begin{align*}|2-4|=?\end{align*}|24|=?
  2. \begin{align*}|5-6|=?\end{align*}|56|=?
  3. \begin{align*}\big | -\frac{1}{2} - \frac{5}{6}\big |=?\end{align*}1256=?
  4. \begin{align*}|2.3-3.7|=?\end{align*}|2.33.7|=?
  5. \begin{align*}|7.8 - 9.4|=?\end{align*}|7.89.4|=?

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Vocabulary

TermDefinition
Absolute Value The absolute value of a number is the distance the number is from zero. Absolute values are never negative.

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MAT.ALG.156.2.L.3