SOLVING RATIONAL ABSOLUTE VALUE EQUATIONS

Problem 1 :

Solve for x :

|(3x + 2)/(1 – x)| = 4

Solution :

Decomposing into two branches, we get

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

Solving the first branch :

(3x + 2)/(1 – x) = 4

Multiply each side by (1 – x).

(3x + 2) = 4(1 – x)

3x + 2 = 4 – 4x

Adding 4x on each sides.

3x + 4x + 2 = 4 – 4x + 4x

7x + 2 = 4

Subtracting 2 on each sides.

7x = 4 – 2

7x = 2

x = 2/7

Solving the second branch :

(3x + 2)/(1 – x) = -4

Multiply each side by (1 – x).

(3x + 2) = -4(1 – x)

3x + 2 = -4 + 4x

Subtracting 4x on each sides.

3x - 4x + 2 = -4 + 4x - 4x

-x + 2 = -4

Subtracting 2 on each sides.

-x = -4 - 2

-x = -6

x = 6

So, the values of x are 2/7 and 6.

Problem 2 :

Solve for x :

|x /(x – 1)| = 3

Solution :

Decomposing into two branches, we get

x /(x – 1) = 3 and x /(x – 1) = -3

x/(x – 1) = 3

Multiply each side by (x – 1).

x = 3(x – 1)

x = 3x – 3

x = 3/2

Solving the second branch :

x/(x – 1) = -3

Multiply each side by (x – 1).

x = -3(x – 1)

x = -3x + 3

Adding 3x on each sides.

x + 3x = 3

4x = 3

Dividing 4 on each sides.

4x/4 = 3/4

x = 3/4 

So, the values of x are 3/2 and 3/4.

Problem 3 :

Solve for x :

|(2x - 1)/(x + 1)| = 5

Solution :

(2x - 1)/(x + 1) = 5

Multiply each side by (x + 1).

2x - 1 = 5(x + 1)

2x - 1 = 5x + 5

Subtracting 2x on each sides.

-1 = 5x – 2x + 5

-1 = 3x + 5

Subtracting 5 on each sides.

-1 – 5 = 3x

-6 = 3x

Dividing 3 on each sides.

-6/3 = 3x/3

-2 = x

Solving for second branch :

(2x - 1)/(x + 1) = -5

Multiply each side by (x + 1).

2x - 1 = -5(x + 1)

2x - 1 = -5x - 5

Adding 5x on each sides.

2x + 5x – 1 = -5

7x – 1 = -5

Adding 1 on each sides.

7x = -5 + 1

7x = -4

Dividing 7 on each sides.

x = -4/7

So, the values of x are -2 and -4/7.

Problem 4 :

Solve for x :

|(x + 3)/(1 – 3x)| = 1/2

Solution :

Decomposing into two branches, we get

(x + 3)/(1 – 3x) = 1/2 and (x + 3)/(1 – 3x) = -1/2

Solving the first branch :

(x + 3)/(1 – 3x) = 1/2

Multiply each side by (1 – 3x).

x + 3 = 1/2)(1 – 3x)

x + 3 = 1/2 – (3/2)x

Adding 3/2 on each sides.

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

x/1 × 2/2 + (3/2)x + 3 = 1/2

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

(5/2)x + 3 = 1/2

Subtracting 3 on each sides.

(5/2)x = 1/2 – 3

(5/2)x = -5/2

x = -1

Solving for second branch :

(x + 3)/(1 – 3x) = -1/2

Multiply each side by (1 – 3x).

x + 3 = -1/2(1 – 3x)

x + 3 = -1/2 – (-3/2)x

Subtracting 3/2 on each sides.

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

x/1 × 2/2 - (3/2)x + 3 = -1/2

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

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

Subtracting 3 on each sides.

(-1/2)x = -1/2 – 3

(-1/2)x = -7/2

Multiplying 2 on each sides.

(-1/2)x × 2 = -7/2 × 2

-x = -7

x = 7

So, the values of x are -1 and 7.

Problem 5 :

Solve for x :

|x/(x – 2)| = 3

Solution :

Decomposing into two branches, we get

x/(x – 2) = 3 and x/(x – 2) = -3

Solving the first branch :

x/(x – 2) = 3

Multiplying (x – 2) on each sides.

(x/(x – 2)) × (x – 2) = 3 × (x – 2)

x = 3x – 6

Subtracting 3x on each sides.

x – 3x = -6

-2x = -6

x = 3

Solving the second branch :

x/(x – 2) = -3

Multiplying (x – 2) on each sides.

(x/(x – 2)) × (x – 2) = -3 × (x – 2)

x = -3x + 6

Subtracting 3x on each sides.

x + 3x = 6

4x = 6

x = 3/2

So, the values of x are 3 and 3/2.

Problem 6 :

Solve for x :

|(2x + 3)/(x – 1)| = 2

Solution :

(2x + 3)/(x – 1) = 2

Multiplying (x – 1) on each sides.

((2x + 3)/(x – 1)) × (x – 1) = 2 × (x – 1)

2x + 3 = 2x – 2

Subtracting 2x on each sides.

2x – 2x = -2 – 3

0 = -5

No solution.

(2x + 3)/(x – 1) = -2

Multiplying (x – 1) on each sides.

((2x + 3)/(x – 1)) × (x – 1) = -2 × (x – 1)

2x + 3 = -2x + 2

Adding 2x on each sides.

2x + 2x + 3 = -2x + 2 + 2x

4x + 3 = 2

4x = 2 – 3

4x = -1

x = -1/4

So, the value of x is -1/4.