# 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

### Solution of Rational Equations

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

In this Concept you will learn how to solve using cross products.

**Solving a Rational Proportion**

When two rational expressions are equal, a **proportion** is created and can be solved using its cross products.

#### Example A

For example, to solve

Solve for

#### Example B

*Solve 2xx+4=5x.*

**Solution:**

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

#### Example C

Solve

**Solution:**

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.

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

2. What factors of -6 add up to 5? That would be -6 and 1, since -6+1=-5.

3. Factor, beginning by breaking up the middle term,

4. Use the Zero Product Principle:

### Video Review

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### Guided Practice

*Solve −x2=3x−8x.*

**Solution:**

### Explore More

Sample explanations for some of the practice exercises below are available by viewing the following videos. Note that there is not always a match between the number of the practice exercise in the videos and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Solving Rational Equations (12:57)

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) .

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 12.8.

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

## Description

## Learning Objectives

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

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## Date Created:

Feb 24, 2012## Last Modified:

Aug 20, 2015## Vocabulary

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