# 6.5: Absolute Value Equations

**At Grade**Created by: CK-12

## Learning Objectives

At the end of this lesson, students will be able to:

- Solve an absolute value equation.
- Analyze solutions to absolute value equations.
- Graph absolute value functions.
- Solve real-world problems using absolute value equations.

## Vocabulary

Terms introduced in this lesson:

- absolute value
- absolute value equations
- vertex or cusp

## Teaching Strategies and Tips

Focus on the interpretation of absolute value as a distance.

- In Example 1b,
|−120|=120 because–120 is120units from the origin. - In Example 2, have students explain,
*using a distance argument*, why the order in which the two numbers are subtracted is not important. In general, for any two numbers (or points)a andb ,|a−b|=|b−a| . - Have students rethink the simple absolute value equations
|x|=3 and|x|=10 in Example 3 as|?|=3 and|?|=10 , respectively. (Which numbers are3units from the origin?10units from the origin?) - Have students interpret absolute value equations out loud.
|x−2|=7 means “those numbers on the number line7units away from2 .” See Examples 4-6. Encourage students to draw the number line and mark the possible solutions. - Using the distance interpretation, point out that absolute value equations (involving only linear functions) can have no more than
2 solutions. Have students consider absolute value equations with1 or solutions such as|x−2|=0 and|x−2|=5 , respectively.

Students have trouble reconciling the definition “

- Let
x=–5 . Then|−5|=−(−5) since–5 is negative (using the definition). This simplifies to|−5|=5 . -
|−5|=5 since absolute value changes a negative number into its positive inverse.

Use Examples 5 and 6 to show students how to rewrite absolute value equations so that the distance interpretation is clearer.

Additional Examples:

a. *Solve the equation and interpret the answer.*

Solution: As it stands, the equation cannot be interpreted in terms of distance. Rewrite the equation with a minus sign:

b. *Solve the equation and interpret the answer.*

Hint: As it stands, the equation cannot be interpreted in terms of distance. Rewrite the equation by dividing both sides by

This last equation can now be interpreted as those numbers

Treat the absolute value as a grouping symbol when appropriate.

- The distributive law holds in expressions such as
3|x−4|=|3x−12| and13|3x−6|=∣∣∣3x3−63∣∣∣=|x−2| . - The distributive law does not hold in an expression such as
−2|x−3|≠|−2x+6| - In general, distribute into absolute value
a⋅|b|=|a⋅b| whena=|a| ; i.e., for positive numbersa . - These steps are based on the property
|a|⋅|b|=|a⋅b| .

When beginning to graph absolute value functions, encourage students to make a table of values such as those in Examples 7 and 8.

Have students plot and describe in words the basic graph

- The graph has a
“V” shape, consisting of two rays that meet at a sharp point, called the vertex or cusp. - One side of the
“V” has positive slope and other side negative slope. - The vertex is located at the point where the expression inside the absolute value is equal to zero.

## Error Troubleshooting

General Tip: Remind students *not* to distribute a negative into the absolute value expression. For example,

General Tip: Students may misinterpret “absolute value is always positive” and commit the error *after* the absolute value has been performed, and so \begin{align*}-|3 - 5| = (-1)|3 - 5| = (-1) |-2| = (-1) 2 = -2\end{align*}.

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