12.8: 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
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:
Guided Practice
Solve
Solution:
Practice
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. CK12 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) .
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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.