Solving Rational Equations

 

Learning Objective(s)

·         Solve rational equations using the techniques for simplifying and manipulating rational expressions.

 

Introduction

 

Equations that contain rational expressions are called rational equations. We can solve these equations using the techniques for performing operations with rational expressions and for solving algebraic equations.

 

Solving Rational Equations Using Common Denominators

 

One method for solving rational equations is to rewrite the rational expressions in terms of a common denominator. Then, since we know the numerators are equal, we can solve for the variable. To illustrate this, let’s look at a very simple equation:

 

 

x = 3

 

Since the denominator of each expression is the same, the numerators must be equivalent as well.  This means that x = 3.

 

This is true for rational equations with polynomials too:

 

 

2x – 5 = 11

 

x = 8

 

Again, since the denominators are the same, we know the numerators must also be equal. So we can set them equal to one another and solve for x.

 

We should check our solution in the original rational expression:

 

 

 

 

The solution checks, and since x = 8 does not result in division by 0, the solution is valid.

 

When the terms in a rational equation have unlike denominators, solving the equation will involve some extra work. Here’s an example:

 

Example

Problem

Solve the equation

 

 

 

 

 

There are no excluded values because the denominators are both constants.

 

 

 

Find a common denominator and rewrite each expression with that denominator.

 

The common denominator is 8.

 

 

x + 2 = 6

 

x = 4

 

Since the denominators are the same, the numerators must be equal for the equation to be true. Solve for x.

 

 

 

 

Check the solution by substituting 4 for x in the original equation.

Answer

 

x = 4

 

 

 

Another way of solving rational equations is to multiply both sides of the equation by the common denominator. This eliminates the denominators and turns the rational equation into a polynomial equation. Here is the same equation we just solved:

 

Example

Problem

Solve the equation

 

 

 

 

 

There are no excluded values because the denominators are both constants.

 

 

 

Multiply both sides by the least common denominator

 

 

 

x + 2 = 6

 

Simplify

 

 

x + 2 – 2  = 6 – 2

 

x = 4

 

Solve for x

Answer

 

x = 4

 

 

 

Now that we understand the techniques, let’s look at an example that has variables in the denominator too. Remember that whenever there are variables in the denominator, we need to find any values that are excluded from the domain because they'd make the denominator zero.

 

To solve this equation, we can multiply both sides by the least common denominator:

 

Example

Problem

Solve

 

 

 

x + 2 = 0

x = -2

 

x – 2 = 0

x = 2

 

(x + 2)(x – 2) = 0

x = -2, 2

 

First determine the excluded values. These are the values of x that result in a 0 denominator.

 

denominators:

 

x + 2

x – 2

x2 – 4 = (x – 2)(x + 2)

 

least common denominator:

(x – 2)(x + 2)

 

 

Find the common denominator of x – 2, x + 2, and x2 – 4

 

Since (x – 2) and (x + 2) are both factors of x2 – 4, the least common denominator is (x – 2)(x + 2) or x2 – 4

 

 

 

 

Multiply both sides of the equation by the common denominator.

 

 

7x – 14 + 5x + 10 =10x – 2

 

12x – 4 =10x – 2

 

 

Simplify

 

 

12x – 10x – 4 = 10x – 10x – 2

 

2x – 4 = -2

 

2x – 4 + 4 = -2 + 4

 

2x = 2

 

x = 1

 

Solve for x

 

 

 

 

 

 

Check to be sure that the solution is not an excluded value. (It is not.)

 

 

 

 

 

 

 

Check the solution in the original equation.

Answer

x = 1

 

 

 

 

Solve the equation , m  0 or 2

 

A)  m = 2

B) no solution

C) m = 8

 

Show/Hide Answer

A) Incorrect.  You probably found the common denominator correctly, but forgot to distribute when you were simplifying. You also forgot to check your solution or note the excluded values; m ≠ 2 because it makes the expression on the right side undefined. Multiplying  both sides by the common denominator gives  , so . The correct answer is m = 8.

 

B) Incorrect. , so .  The solution, 8, is not an excluded value. The correct answer is m = 8.

 

C) Correct. Multiplying both sides of the equation by the common denominator gives , so . . The correct answer is m = 8.

 

 

We've seen that there is more than one way to solve rational equations. Because both of these techniques manipulate and rewrite terms, sometimes they can produce solutions that don't work in the original form of the equation. These types of answers are called extraneous solutions. These solutions are artifacts of the solving process and not real answers at all. That's why we should always check solutions in the original equations—we may find that they yield untrue statements or produce undefined expressions.

 

Solve the equation:

 

A) x = -1

B) x = -1, 6

C) x = -4, 3

D) no solution

 

Show/Hide Answer

A) Correct. ; ; x – 2 + x2 – 6x = 4; (x – 6 )(x + 1) = 0. Since 6 is an excluded value, it is an extraneous solution. Only -1 is a real solution. 

 

B) Incorrect. 6 is an excluded value because it makes the denominator of the first rational expression equal to 0. Since 6 is an extraneous solution, it can't be included in the solution.  The correct answer is -1.

 

C) Incorrect. The common denominator is (x – 6 )(x -2). Each term on the left side must be multiplied by a fraction equivalent to 1 that will produce that denominator: = . The correct answer is -1.

 

D) Incorrect. When the equation is solved by finding the common denominator, the answers are -1 and 6. It is true that 6 is an excluded value and thus an extraneous solution that must be discarded. But -1 works in the original equation and it is a valid solution.  The correct answer is -1.

 

Summary

 

We solve rational equations by finding a common denominator. We can then follow either of two methods. We can rewrite the equation so that all terms have the common denominator and we can solve for the variable with just the numerators. Or we can multiply both sides of the equation by the common denominator so that all terms become polynomials instead of rational expressions.

 

An important step in solving rational equations is to reject any extraneous solutions from the final answer. Extraneous solutions are solutions that don't satisfy the original form of the equation because they produce untrue statements or are excluded values that make a denominator equal to 0.