SOLVING ABSOLUTE VALUE EQUATIONS

About "Solving Absolute Value Equations"

Solving Absolute Value Equations : 

In this section, we are going to learn, how to solve absolute value equations. 

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)

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

We can solve the absolute value equation |2x + 3| = 5 as shown below. 

Steps Involved in Solving 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. 

Some More 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

Hence the solution is x  =  -4, 2/3.

Example 2 :

Solve the absolute value equation :

|7x|  =  21

Solution :

Solve :

|7x|  =  21

7x  =  21     or     7x  =  -21

x  =  3     or     x  =  -3

Hence, the solution is x  =  -3, 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

Hence, the solution is x  =  -2, -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

Hence, the solution is x  =  3, 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

Hence the solution is x  =  -1/6, -5/2.

Example 6 :

Solve the absolute value equation :

3|5x - 6| - 4  =  5

Solution :

Solve : 

3|5x - 6| - 4  =  5

Add 4 to each side.

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

Hence the solution is x  =  9/5, 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.

Hence, 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

Add 0.5 to each side. 

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

Hence the solution is x  =  -12, 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

Hence, the value of k  =  1, 7.

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

Hence the value of k  =  8.

After having gone through the stuff given above, we hope that the students would have understood, "Solving Absolute Value Equations". 

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