# 12.7: Solutions of Rational Equations

## Learning Objectives

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

- Solve rational equations using cross products.
- Solve rational equations using lowest common denominators.
- Solve real-world problems with rational equations.

## Vocabulary

Terms introduced in this lesson:

- rational equation
- cross products

## Teaching Strategies and Tips

Students should now be able to solve linear, quadratic, and radical equations; and be able to graph linear, quadratic, exponential, and radical functions. In this lesson, rational equations and functions are added to these lists.

Emphasize that the first step in solving rational equations is eliminating the denominators on all the terms.

- Emphasize that the last step is checking their answers in the original equation.

Use Example 1 to remind students about cross-multiplication.

- Suggest that students simplify the terms of the equation and move them to one side before cross-multiplying.

Use Example 4 to motivate using the LCD strategy.

- Students learn to eliminate denominators by multiplying all terms by the LCD.
- Emphasize that this method is preferred over cross-multiplication when there are two or more terms in an equation.
- Have students completely factor each expression first.

After multiplying both sides by the LCD, have students use colors to cancel like factors. This helps students track their canceling and prevents canceling a factor more than once.

Reconstruct the tables in Examples 7-9 as a useful way to display the given information in *Review Questions* 19-24.

## Error Troubleshooting

General Tip: Remind students to check their answers in the *original* equation.

- Emphasize that this is a necessary step for rational equations since the variable appears in denominators.
- Essentially, if an answer makes any denominator zero, then that value is not a solution to the equation.