SOLVING RATIONAL EQUATIONS

Solve for x :

Problem 1 :

(2x + 3)/(x + 1) = 4/3

Solution :

(2x + 3)/(x + 1) = 4/3

Doing cross multiplication, we get

3(2x + 3) = 4(x + 1)

Using distributive property, we get

6x + 9 = 4x + 4

Subtracting 4x and 9 on both sides.

6x – 4x = 4 – 9

2x = -5

Dividing by 2 on both sides.

x = -5/2

Problem 2 :

(x + 1)/(1 – 2x) = 3/5

Solution :

(x + 1)/(1 – 2x) = 3/5

Doing cross multiplication, we get

5(x + 1) = 3(1 – 2x)

Using distributive property, we get

5x + 5 = 3 – 6x

Adding 6x and subtracting 5 on both sides, we get

5x + 6x = 3 – 5

11x = -2

x = -2/11 

Problem 3 :

(2x – 1)/(4 – 3x) = -2/3

Solution :

(2x – 1)/(4 – 3x) = -2/3

Doing cross multiplication, we get

3(2x -1) = -2(4 – 3x)

Using distributive property, we get

6x – 3 = -8 + 6x

Subtracting 6x on both sides and adding 3 on both side.

6x – 6x = -8 + 3

0 = -5

It is not true. So, no solution.

Problem 4 :

(x + 3)/(2x – 1) = 1/2

Solution :

(x + 3)/(2x – 1) = 1/2

Doing cross multiplication, we get

2(x + 3) = 2x – 1

Using distributive property, we get

2x + 6 = 2x -1

2x – 2x = -1 – 6

0 = -7

It is not true. So, no solution.

Problem 5 :

(4x + 3)/(2x – 1) = 6

Solution :

(4x + 3)/(2x – 1) = 6

We can consider 6 as 6/1.

(4x + 3)/(2x – 1) = 6/1

Doing cross multiplication, we get

4x + 3 = 6(2x – 1)

By distributing 6.

4x + 3 = 12x – 6

Subtracting 12x and subtracting 3 on both sides.

4x – 12x = -6 – 3

-8x = -9

x = 9/8

Problem 6 :

(3x – 2)/(x + 4) = -5

Solution :

(3x – 2)/(x + 4) = -5

We can consider -5 as -5/1

(3x – 2)/(x + 4) = -5/1

3x – 2 = -5(x + 4)

Distributing -5.

3x – 2 = -5x – 20

Adding 5x and 2 on both sides.

3x + 5x = -20 + 2

8x = -18

x = -18/8

x = -9/4

Problem 7 :

(6x – 1)/(3 – 2x) = 10

Solution :

(6x – 1)/(3 – 2x) = 10

Doing cross multiplication, we get

6x – 1 = 10(3 - 2x)

Distributing 10, we get

6x – 1 = 30 - 20x

Adding 20x and 1 on both sides, we get

6x + 20x = 30 + 1

26x = 31

x = 31/26

Problem 8 :

(5x – 1)/(x + 4) = 5

Solution :

(5x – 1)/(x + 4) = 5

Doing cross multiplication, we get

5x – 1 = 5(x + 4)

5x – 1 = 5x + 20

5x - 5x = 20 + 1

0 = 21

It is not true. So, no solution.