# 12.8: Rational Equations Using Proportions

**Basic**Created by: CK-12

**Practice**Rational Equations Using Proportions

Suppose you were traveling on a paddle boat at a constant speed. In 6 minutes, you traveled

### Rational Equations Using Proportions

You are now ready to solve rational equations! There are two main methods you will learn to solve rational equations:

- Cross-products
- Lowest common denominators

This Concept will focus on cross-products. When two rational expressions are equal, a **proportion** is created and can be solved using its cross-products.

For example, to solve

Solve for

Let's solve the following rational equations:

2xx+4=5x .

Notice that this equation has a degree of two; that is, it is a quadratic equation. We can solve it using the quadratic formula.

3x5x+2=1x

Start by cross multiplying:

Since this equation has a squared term as its highest power, it is a quadratic equation. We can solve this by using the quadratic formula, or by factoring.

Since there are no common factors, start by finding the product of the coefficient in front of the squared term and the constant:

What factors of -6 add up to 5?

That would be -6 and 1, since -6+1=-5

Factor, beginning by breaking up the middle term,

Use the Zero Product Principle:

### Examples

#### Example 1

Earlier, were asked to find the value of

You can use cross-products to find the value of

First, let's set up an equation using the given information. The distance traveled in a specific amount of time can be written as a fraction of the distance over the time.

Now, we can cross multiply.

#### Example 2

Solve* −x2=3x−8x.*

### Review

Solve the following equations.

2x+14=x−310 4xx+2=59 53x−4=2x+1 7xx−5=x+3x 2x+3−1x+4=0 3x2+2x−1x2−1=−2

**Mixed Review**

- Divide:
−2910÷−158 . - Solve for
g:−1.5(−345+g)=20120 . - Find the discriminant of
6x2+3x+4=0 and determine the nature of the roots. - Simplify
6b2b+2+3 . - Simplify
82x−4−5xx−5 . - Divide:
(7x2+16x−10)÷(x+3) . - Simplify
(n−1)∗(3n+2)(n−4) .

### Review (Answers)

To see the Review answers, open this PDF file and look for section 12.8.

### Notes/Highlights Having trouble? Report an issue.

Color | Highlighted Text | Notes | |
---|---|---|---|

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Term | Definition |
---|---|

proportion |
A statement in which two fractions are equal: . |

Zero Product Property |
The only way a product is zero is if one or more of the terms are equal to zero: |

Rational Expression |
A rational expression is a fraction with polynomials in the numerator and the denominator. |

### Image Attributions

Here you'll learn how to use proportions to find the solutions to rational equations.

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