# SOLVING ABSOLUTE VALUE EQUATIONS

The general form of an absolute value equation is

|ax + b| = k

In the above absolute value equation, we can notice that there is only absolute part on the left side.

(Here 'a' and 'k' are real numbers and k  0)

Let us consider the absolute value equation |2x + 3|  =  5.

We can solve the absolute value equation |2x + 3|  =  5 as shown below. The following steps will be useful to solve absolute value equations.

Step 1 :

Get rid of absolute sign and divide it into two branches.

Step 2 :

For the first branch, take the sign as it is on the right side.

Step 3 :

For the second branch, change the sign on the right side.

Step 4 :

Then solve both the branches.

## Examples

Example 1 :

Solve the absolute value equation :

|3x + 5|  =  7

Solution :

Solve :

|3x + 5|  =  7

3x + 5  =  7     or     3x + 5  =  -7

3x  =  2     or     3x  =  -12

x  =  2/3     or     x  =  -4

Example 2 :

Solve the absolute value equation :

|7x|  =  21

Solution :

Solve :

|7x|  =  21

7x  =  21     or     7x  =  -21

x  =  3     or     x  =  -3

Example 3 :

Solve the absolute value equation :

|2x + 5| + 6  =  7

Solution :

Solve :

|2x + 5| + 6  =  7

Subtract 6 from each side.

|2x + 5|  =  1

2x + 5  =  1     or     2x + 5  =  1

2x  =  -4     or     2x  =  -6

x  =  -2     or     x  =  -3

Example 4 :

Solve the absolute value equation :

|x - 3| + 6  =  6

Solution :

Solve :

|x - 3| + 6  =  6

Subtract 6 from each side.

|x - 3| + 6  =  6

Subtract 6 from each side.

|x - 3|  =  0

x - 3  =  0     or     x - 3  =  0

x  =  3     or     x  =  3

Example 5 :

Solve the absolute value equation :

2|3x +4|  =  7

Solution :

Solve :

2|3x +4|  =  7

Divide each side by 2.

|3x + 4|  =  7/2

3x + 4  =  7/2     or     3x + 4  =  -7/2

3x  =  7/2 - 4     or     3x  =  -7/2 - 4

3x  =  -1/2     or     3x  =  -15/2

x  =  -1/6     or     x  =  -15/6

x  =  -1/6     or     x  =  -5/2

Example 6 :

Solve the absolute value equation :

3|5x - 6| - 4  =  5

Solution :

Solve :

3|5x - 6| - 4  =  5

3|5x - 6|  =  9

Divide each side by 3.

|5x - 6|  =  3

5x - 6  =  3     or     5x - 6  =  -3

5x  =  9     or     5x  =  3

x  =  9/5     or     x  =  3/5

Example 7 :

Solve the absolute value equation :

|x² - 4x - 5| = 7

Solution :

Solve :

|x2 - 4x - 5|  =  7

x2 - 4x  - 5  =  7     or     x2 - 4x - 5  =  -7

x2 - 4x  - 12  =  0     or     x2 - 4x + 2  =  0

Solve the first quadratic equation x² - 4x - 12  =  0.

x2 - 4x  - 12  =  0

(x + 2)(x - 6)  =  0

x + 2  =  0     or     x - 6  =  0

x  =  -2     or     x  =  6

Solve the second quadratic equation x² - 4x + 2  =  0.

This quadratic equation can not be solved using factoring. Because the left side part can not be factored.

So, we can use quadratic formula and solve the equation as shown below. So, the solution is x  =  -2, 7, 2 ± √2.

Example 8 :

Solve the absolute value equation :

0.5|0.5x| - 0.5  =  2.5

Solution :

Solve :

0.5|0.5x| - 0.5  =  2.5

0.5|0.5x|  =  3

Divide each side by 0.5

|0.5x|  =  6

0.5x  =  6     or     0.5x  =  -6

x  =  12     or     x  =  -12

Example 9 :

If the absolute value equation |2x + k|  =  3 has the solution x  =  -2, find the value of k.

Solution :

Because x  =  -2 is a solution, substitute x  =  -2 in the given absolute value equation.

|2(-2) + k|  =  3

|-4 + k|  =  3

Solve for k :

-4 + k  =  3     or     -4 + k  =  -3

k  =  7     or     k  =  1

Example 10 :

If the absolute value equation |x - 3| - k = 0 has the solution x  =  -5, find the value of k.

Solution :

Because x  =  -2 is a solution, substitute x  =  -2 in the given absolute value equation.

|-5 - 3| - k  =  0

|-8| - k  =  0

8 - k  =  0

8  =  k Apart from the stuff given aboveif you need any other stuff in math, please use our google custom search here.

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