SOLVING RATIONAL EQUATIONS

Solve each of the following rational equations for x :

Example 1 :

Solution :

In this equation, there is one restriction :

x ≠ 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 :

x ≠ 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 :

x ≠ -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 :

x ≠ -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 :

x ≠ 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 :

x ≠ 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 :

x ≠ -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 :

x ≠ -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

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

Already, we know that there is a restriction x ≠ -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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