SOLVING RATIONAL EQUATIONS

Rational equation is an equation involving rational expressions. It can be written in the form,

(where f(x) and g(x) are polynomial functions with no common factors and g(x) ≠ 0. The zeros of f(x) are the solutions of the equation)

Rational expressions typically contain a variable in the denominator. For this reason, we will take care to ensure that the denominator is not 0 by making note of restrictions and checking our solutions.

For some value of x, say x = a, if one or more denominators in the rational equation becomes zero, then there is a restriction ≠ a. When solving the equation, even if you get a solution x = a, it can not be considered as a solution to the given rational equation.

To solve rational equations, you can clear the fractions by getting rid of the denominators. To get rid of the denominators one by one, multiply both sides of the equation by the expression in the denominator. To know more about this, please go over the examples given below. 

Note :

If you find only one fraction on each side of the rational equation, the equation can be solved by cross multiplication. That is, multiply numerator on the left side by denominator on the right side and multiply numerator on the right side by denominator on the left side.

Solve each of the following rational equations for x :

Example 1 :

Solution :

In this equation, there is one restriction :

≠ 0

Least common multiple of the denominators x and 3 is 3x.

Multiply each side of the above equation by 3x to get rid of the denominators. 

15 - x = 3

Subtract 15 from each side. 

-x = -12

Multiply each side by -1.

x = 12

Example 2 :

Solution :

In this equation, there are two restrictions :

≠ 0 and x ≠ -4

Least common multiple of the denominators x and (x + 4) is x(x + 4).

Multiply each side of the above equation by x(x + 4) get rid of the denominators. 

Distribute. 

3(x + 4)  =  2x

3x + 12  =  2x

Subtract 2x from each side. 

x + 12  =  0

Subtract 12 from each side.

x  =  -12

Example 3 :

Solution :

In this equation, there is one restriction :

≠ -2

We have the same denominator on both sides of the equation, that is (x + 2). 

Multiply each side of the equation by (x + 2) to get rid of the denominator. 

3 + 5x + 10  =  4

5x + 13  =  4

Subtract 13 from each side. 

5x =  -9

Divide each side by 5.

x  =  -9/5

Example 4 :

Solution :

In this equation, there is one restriction :

≠ -4

We have the same denominator on both sides of the equation, that is (x + 4). 

Multiply each side of the equation by (x + 2) to get rid of the denominator. 

2x - 3x - 12  =  -12

-x - 12  =  -12

Add 12 to each side. 

-x  =  0

x  =  0

Example 5 :

Solution :

In this equation, there are two restrictions :

≠ 2 and x ≠ 0

Least common multiple of the denominators (2 -x) and x is x(2 - x).

Multiply each side of the above equation by x(2 - x) get rid of the denominators. 

14x  =  2(2 - x)

14x  =  4 - 2x

Add 2x to each side. 

16x  =  4

Divide each side by 16. 

x  =  1/16

Example 6 :

Solution :

In this equation, there is one restriction :

≠ 4

We have the same denominator on both sides of the equation, that is (x - 4). 

Multiply each side of the equation by (x - 2) to get rid of the denominator. 

2 + 2x - 8  =  6

2x - 6  =  6

Add 6 to each side. 

2x  =  12

Divide each side of the equation by 2.

x  =  6

Example 7 :

Solution :

In this equation, there is one restriction :

≠ -1

We have the same denominator on both sides of the equation, that is (x + 1). 

Multiply each side of the equation by (x + 1) to get rid of the denominator. 

x + 2 - x2 - x  =  -6

2 - x2  =  -6

Subtract 2 from each side. 

-x2  =  -8

Multiply each side by -1.

x2  =  8

Take square root on both sides.

√x2  =  √8

x  =  ± √(2 ⋅ 2 ⋅ 2)

x  =  ± 2√2

x  =  -2√2  or  2√2

Example 8 :

Solution :

In this equation, there are two restrictions :

≠ -2 and x ≠ -3

Least common multiple of the denominators is (x+2)(x+3).

Multiply each side of the above equation by (x + 2)(x + 3) to get rid of the denominators.

Use the distributive property and simplify. 

x(x + 3) + 2  =  5(x + 2)

x2 + 3x + 2  =  5x + 10

Subtract 5x and 10 from each side. 

x2 - 2x - 8  =  0

Multiply the coefficient of x2, that is 1 and the constant -8. The result is -8. Find two factors for -8 such that the product is -8 and sum is equal to the coefficient of x, that is -2. Then, the two factors are -4 and +2.

x2 - 4x + 2x - 8  =  0

x(x - 4) + 2(x - 4)  =  0

(x - 4)(x + 2)  =  0

x - 4 = 0

x = 4

x + 2 = 0

x = -2

We get two solutions x = 4 and x = -2 for the given rational equation.

Already, we know that there is a restriction ≠ -2. 

So, x = -2 can not be considered as a solution.

Therefore, solution to the given rational equation is

x = 4

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