SOLVING ABSOLUTE VALUE EQUATIONS WITH ABSOLUTE SIGN ON BOTH SIDES

Solve the following absolute value equations.

Problem 2 :

|x| = |5 – x|

Solution :

|x| = |5 – x|

Decomposing into two branches, we get

x = (5 – x)    (or)      x = -(5 – x)

Solving for x = 5 – x :

Adding x on each sides.

x + x = 5 – x + x 

2x = 5

Dividing 2 on each sides.

2x/2 = 5/2

x = 5/2

Solving for x = -(5 – x) :

x = -(5 – x)

x = -5 + x

Subtracting x on each sides.

x – x = -5 + x - x

0 = -5

Problem 3 :

|3x - 1| = |x + 2|

Solution :

|3x - 1| = |x + 2|

Decomposing into two branches, we get

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

Subtracting x on each sides.

3x - 1 – x = x + 2 - x

2x – 1 = 2

Adding 1 on each sides.

2x – 1 + 1 = 2 + 1

2x = 3

Dividing 2 on each sides.

2x/2 = 3/3

x = 3/2

(or)

3x - 1 = -(x + 2)

3x – 1 = -x – 2

Adding x on each sides.

3x – 1 + x = -x – 2 + x

4x – 1 = -2

Adding 1 on each sides.

4x – 1 + 1 = -2 + 1

4x = -1

Dividing 4 on each sides.

4x/4 = -1/4

x = -1/4 

Problem 4 :

|2x + 5| = |1 – x|

Solution :

|2x + 5| = |1 - x|

Decomposing into two branches, we get

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

Solving for 2x + 5 = 1 – x :

Adding x on each sides.

2x + x + 5 = 1 – x + x

3x + 5 = 1

Subtracting 5 on each sides.

3x + 5 – 5 = 1 - 5

3x = -4

Dividing 3 on each sides.

3x/3 = -4/3

x = -4/3

Solving for 2x + 5 = -(1 - x) :

2x + 5 = -(1 - x)

2x + 5 = -1 + x

Subtracting x on each sides.

2x + 5 - x = -1 + x – x

x + 5 = -1

Subtracting 5 on each sides.

x + 5 - 5 = -1 - 5

x = -6

Both the answers x = -4/3 and x = -6 are correct and acceptable.

Problem 5 :

|1 – 4x| = 2|x – 1|

Solution :

|1 – 4x| = 2|x – 1|

Decomposing the given function into two branches, we get

1 – 4x = 2(x – 1)    (or)     1 – 4x = -2(x – 1)

Solving for 1 – 4x = 2(x – 1) :

1 - 4x = 2x – 2

Subtracting 2x on each sides.

1 - 4x – 2x = 2x – 2 – 2x

1 – 6x = -2

Subtracting 1 on each sides.

1 – 1 - 6x = -2 – 1

-6x = -3

Dividing -6 on each sides.

-6x/-6 = -3/-6

x = 3/6

x = 1/2

Solving for 1 – 4x = -2(x – 1) :

1 – 4x = -2(x – 1)

1 – 4x = -2x + 2

Subtracting 2 on each sides.

1 – 4x - 2 = -2x + 2 – 2

-4x – 1 = -2x

Adding 1 on each sides.

-4x – 1 = -2x

-4x – 1 + 1 = -2x + 1

-4x = -2x + 1

Adding 2x on each sides.

-4x + 2x = -2x + 1 + 2x

-2x = 1

Dividing 2 on each sides.

-2x/2 = 1/2

x = -1/2

Problem 6 :

|3x + 2| = 2|2 – x|

Solution :

|3x + 2| = 2|2 – x|

Decomposing into two branches, we get

(3x + 2) = 2(2 – x)    (or)   (3x + 2) = -2(2 – x)

Solving for (3x + 2) = 2(2 – x) :

3x + 2 = 2(2 – x)

3x + 2 = 4 – 2x

Adding 2x on each sides.

3x + 2 + 2x = 4 – 2x + 2x

5x + 2 = 4

Subtracting 2 on each sides.

5x + 2 – 2 = 4 – 2

5x = 2

Dividing 5 on each sides.

5x/5 = 2/5

x = 2/5

Solving for (3x + 2) = -2(2 – x) :

3x + 2 = -2(2 – x)

3x + 2 = -4 + 2x

Subtracting 2x on each sides.

3x + 2 – 2x = -4 + 2x – 2x

3x + 2 – 2x = -4

Subtracting 2 on each sides.

3x + 2 – 2x – 2 = -4 - 2

x = -6