A scale model of a racecar is in the ratio of 1:
*
x
*
to the real racecar. The length of the model is
units, and the length of the real racecar is
units. What is the value of
*
x
*
?

### Guidance

A
**
rational equation
**
is an equation where there are rational expressions on both sides of the equal sign. One way to solve rational equations is to use cross-multiplication. Here is an example of a proportion that we can solve using cross-multiplication.

If you need more of a review of cross-multiplication, see the
*
Proportion Properties
*
concept. Otherwise, we will start solving rational equations using cross-multiplication.

#### Example A

Solve .

**
Solution:
**
Use cross-multiplication to solve the problem. You can use the example above as a guideline.

Check your answers. It is possible to get extraneous solutions with rational expressions.

#### Example B

Solve .

**
Solution:
**
Cross-multiply and solve.

Check your answers.

and

#### Example C

Solve .

**
Solution:
**
Cross-multiply.

Check the answer:

**
Intro Problem Revisit
**
We need to set up a rational equation and solve for
*
x
*
.

Now cross-multiply.

However,
*
x
*
is a ratio so it must be greater than 0. Therefore
*
x
*
equals 21 and the model is in the ratio 1:21 to the real racecar.

### Guided Practice

Solve the following rational equations.

1.

2.

3.

#### Answers

1.

2.

3.

is not actually a solution because it is a vertical asymptote for each rational expression, if graphed. Because zero is not part of the domain, it cannot be a solution, and is extraneous.

### Vocabulary

- Rational Equation
- An equation where there are rational expressions on both sides of the equal sign.

### Problem Set

- Is a solution to ?

Solve the following rational equations.

Determine the values of
*
a
*
that make each statement true. If there no values, write
*
none.
*

- , such that there is no solution.
- , such that there is no solution.
- , such that there is one solution.
- , such that there are two integer solutions.