HOW TO SOLVE RATIONAL EQUATIONS

What is a rational equation ?

A rational equation is an equation that has one or more rational expressions or more simply put, it's an equation with fractions.

And solving equations with rational expressions can be using two different methods.

Method 1 :

Step 1 :

Simplify both sides of the equation by adding/subtracting fractions to get one fraction on either side. 

Step 2 :

Now, we use the Cross multiplication.

Step 3 :

Solve the transformed equation.

Method 2 :

Step 1 :

Multiply both sides of the equation by the Least common denominator to get rid of the fractions.

Step 2 :

Solve the transformed equation.

Solve and check the following rational equations.

Example 1 :

Solution :

By cross multiplication, we get

12 + 3x  =  9x

12  =  9x - 3x

12  =  6x

x  =  2

Cross-check :

By applying x  =  2 in the given question, we get

4/[3(2)] + 1/3  =  1

2/3 + 1/3  =  1

1  =  1 (true)

So, the solution of x is 2.

Example 2 :

Solution :

By cross multiplication, we get

5a(2)  =  5(a + 4)

10a  =  5a + 20

5a  =  20

a  =  4

Cross-check :

By applying a  =  4 in the given question, we get

[5(4)]/(4 + 4)  =  5/2

20/8  =  5/2

5/2  =  5/2 (true)

So, the solution of a is 4.

Example 3 :

Solution :

By cross multiplication, we get

3(3x + 12)  =  5(x + 4)

9x + 36  =  5x + 20

9x - 5x  =  20 - 36

4x  =  -16

x  =  -4

Cross-check :

By applying x  =  -4 in the given question, we get

[3(-4) + 12]/(-4 + 4)  =  5/3

0/0  =  5/3 (false)

So, there is no solution for x.

Example 4 :

Solution :

By cross multiplication, we get

18m  =  72

m  =  4

Cross-check :

By applying m  =  4 in the given question, we get

1/6 + 1/12  =  1/4

18/72  =  1/4

1/4  =  1/4 (true)

So, the solution of m is 4.

Example 5 :

Solution :

By cross multiplication, we get

x(x + 6)  =  8(x + 3)

x2 + 6x  =  8x + 24

x+ 6x - 8x - 24  =  0

x2 - 2x - 24  =  0

By factorization, we get

(x - 6)(x + 4)  =  0

x - 6  =  0 and x + 4  =  0

x  =  6 and -4

Cross-check :

By applying x  =  6 in the given question, we get

6/(6 + 3)  =  8/(6 + 6)

6/9  =  8/12

2/3  =  2/3 (true)

By applying x  =  -4 in the given question, we get

-4/(-4 + 3)  =  8/(-4 + 6)

-4/(-1)  =  8/2

4/1  =  4/1 (true)

So, the solution of x is {6, -4}

Example 6 :

Solution :

By cross multiplication, we get

(b - 2)(b - 5)  =  10

b2 - 5b - 2b + 10  =  10

b2 - 7b + 10 - 10  =  0

b2 - 7b  =  0

b(b - 7)  =  0

b  =  0 and b  =  7

Cross-check :

By applying b  =  0 in the given question, we get

(0 - 2)/2  =  5/(0 - 5)

(-2)/2  =  5/(-5)

-1  =  -1 (true)

By applying b  =  7 in the given question, we get

(7 - 2)/2  =  5/(7 - 5)

5/2  =  5/2 (true)

So, the solution of b is {0, 7}

Example 7 :

Solution :

By cross multiplication, we get

(x + 3)(x + 3)  =  2x + 5

x2 + 3x + 3x + 9  =  2x + 5

x+ 6x + 9 - 2x - 5  =  0

x+ 4x + 4  =  0

(x + 2)(x + 2)  =  0

x  =  -2, -2

Cross-check :

By applying x  =  -2 in the given question, we get

(-2 + 3)/[2(-2) + 5]  =  1/(-2 + 3)

1/1  =  1/1 (true)

So, the solution of x is -2.

Example 8 :

Solution :

By cross multiplication, we get

(y + 1)(y + 5)  =  (y + 4)(y - 1)

y2 + 5y + y + 5  =  y2 - y + 4y - 4

y+ 6y + 5  =  y+ 3y - 4

y+ 6y + 5 - y2 - 3y + 4  =  0

3y + 9  =  0

y  =  -3

Cross-check :

By applying y  =  -3 in the given question, we get

(-3 + 1)/(-3 - 1)  =  (-3 + 4)/(-3 + 5)

(-2)/(-4)  =  1/2

1/2  =  1/2 (true)

So, the solution of y is -3.

Example 9 :

Solution :

By cross multiplication, we get

k2 - 2k  =  6(k - 2)

k- 2k  =  6k - 12

k- 2k - 6k + 12  =  0

k- 8k + 12  =  0

(k - 6)(k - 2)  =  0

k  =  6 and 2

Cross-check :

By applying k  =  6 in the given question, we get

1/(6 - 2)  =  6/[62 - 2(6)]

1/4  =  6/24

1/4  =  1/4 (true)

By applying k  =  2 in the given question, we get

1/(2 - 2)  =  6/[2- 2(2)]

1/0  =  6/0

=  infinity

Since it has an infinity, we can't take the k  =  2.

So, the required solution of k is 6.

Example 10 :

Solution :

By cross multiplication, we get

(-x + 14)(x - 2)  =  5(3x - 6)

-x2 + 2x + 14x - 28  =  15x - 30

-x2 + 16x - 28  =  15x - 30

-x2 + 16x - 28 - 15x + 30  =  0

-x+ x + 2  =  0

-(x2 - x - 2)  =  0

x- x - 2  =  0

(x - 2)(x + 1)  =  0

x  =  2 and -1

Cross-check :

By applying x  =  2 in the given question, we get

(2 + 2)/(2 - 2) - 4/3  =  5/(2 - 2)

4/0 - 4/3  =  5/0

=  infinity

Since it has an infinity, we can't take the k  =  2.

By applying x  =  -1 in the given question, we get

(-1 + 2)/(-1 - 2) - 4/3  =  5/(-1 - 2)

(-1/3) - 4/3  =  (-5/3)

-5/3  =  -5/3 (true)

So, the required solution of x is -1.

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