# 6.5: Absolute Value Equations

## 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, because is 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) and , . - Have students rethink the simple absolute value equations and in Example 3 as and , respectively. (Which numbers are from the origin? from the origin?)
- Have students interpret absolute value equations out loud. means “those numbers on the number line away from .” 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 solutions. Have students consider absolute value equations with or solutions such as and , respectively.

Students have trouble reconciling the definition “ if is negative” and the fact that “absolute value changes a negative number into its positive inverse.” Offer an example:

- Let . Then since is negative (using the definition). This simplifies to .
- 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: which can now be interpreted as those numbers away from . Therefore, the solution set is .

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 away from .

Treat the absolute value as a grouping symbol when appropriate.

- The distributive law holds in expressions such as and .
- The distributive law does not hold in an expression such as
- In general, distribute into absolute value when ; i.e., for positive numbers .
- These steps are based on the property .

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 . Allow them to observe the essential properties of the absolute value graph:

- The graph has a shape, consisting of two rays that meet at a sharp point, called the vertex or cusp.
- One side of the 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 . Suggest to students that in such situations . The multiplication by happens *after* the absolute value has been performed, and so .