# SOLVE ABSOLUTE VALUE EQUATIONS

Solve Absolute Value Equations :

To solve any absolute value function, it has to be in the form of

|x + a|  =  k

Here, a and k are real numbers. And there should be only absolute part on the left side.

Let us consider the absolute value equation given below.

|2x + 3|  =  5 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.

## Solve Absolute Value Equations - Examples

Example 1 :

Solve :

|6m|  =  42 6m  =  42Divide each side by 6.m  =  7 6m  =  42Divide each side by 6. m  =  -7

Hence, the solution is {7, -7}.

Example 2 :

Solve :

|6x|  =  30 6x  =  30Divide each side by 6. x  =  5 6x  =  -30Divide each side by 6. x  =  -5

Hence, the solution is {5, -5}.

Example 3 :

Solve :

|k - 10|  =  3 k - 10  =  3Add 10 to each side.k  =  13 k - 10  =  -3Add 10 to each side.k  =  7

Hence, the solution is {13, 7 }

Example 4 :

Solve :

|x/7|  =  3 x/7  =  3Multiply each side by 7.x  =  21 x/7  =  -3Multiply each side by 7.x  =  -21

Hence the solution is {21, -21}

Example 5 :

Solve :

|a - 5|/8  =  5

Solution :

|a - 5|/8  =  5

Multiply each side by 8.

|a - 5|  =  40

 a - 5  =  40Add 5 to each side.a  =  45 a - 5  =  -40Add 5 to each side.a  =  -35

Hence, the solution is {45, -35}

Example 6 :

Solve :

-3|P|  =  -12

Solution :

-3|P|  =  -12

Divide each side by -3.

|P|  =  4

 p  =  4 p  =  -4

Hence the solution is {4, -4}

Example 7 :

Solve :

|7m| + 3  = 73

Solution :

|7m| + 3  = 73

Subtract 3 from each side.

|7m|  =  70

 7m  =  70Divide each side by 7. m  =  10 7m  =  -70Divide each side by 7. m  =  -10

Hence, the solution is {10, -10}

Example 8 :

Solve :

-10|v + 2|  =  -70

Solution :

-10|v + 2|  =  -70

Divide each side by -10.

|v + 2|  =  7

 v + 2  =  7Subtract 2 from each side. v  =  5 v + 2  =  -7Subtract 2 from each side.v  =  -9

Hence, the solution is {-9, 5}.

Example 9 :

Solve :

|-9 + v|/8  =  3

Solution :

|-9 + v|/8  =  3

Multiply each side by 8.

|-9 + v|  =  24

 -9 + v  =  24Add 9 to each side. v  =  33 -9 + v  =  -24Add 9 to each side.  v  =  -15

Hence, the solution is {33, -15}.

Example 10 :

Solve :

|n| + 1  =  2

Solution :

|n| + 1 = 2

Subtract 1 on both sides.

|n|  =  1

 n  =  1 n  =  -1

Hence, the solution is {1, -1}. After having gone through the stuff given above, we hope that the students would have understood, how to solve absolute value equations.

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