SOLVING ABSOLUTE VALUE EQUATIONS

"Solving absolute value equations" is the stuff which is being studied by the students who study high school math.

Here we are going to see, "How to solve an absolute value equation"

And do remember, before solving any absolute value function, it has to be in the form of |x + a| = k (There should be only absolute part on the left side)

(Here "a" and "k" are real numbers)

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

The picture given below clearly explains, "How the above absolute value equation can be solved".

Steps involved

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 function |3x + 5| = 7

Solution :

When we apply the method explained above for |3x + 5| = 7,

we get  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 function |7x| = 21

Solution :

When we apply the method explained above for |7x| = 21,

we get  7x = 21   or    7x = -21

x = 3    or     x = -3

Hence the solution is x = -3, 3

Example 3 :

Solve the absolute value function |2x + 5| + 6  = 7

Solution :

Let us write the given absolute value equation in the form

|x+a| = k

|2x + 5| + 6  = 7 ------------> |2x+5| = 1

When we apply the method explained above for |2x + 5| = 1,

we get  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 function |x - 3| + 6  = 6

Solution :

Let us write the given absolute value equation in the form

|x+a| = k

|x - 3| + 6  = 6 ------------> |x - 3| = 0

When we apply the method explained above for |x - 3| = 0,

we get  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 function 2|3x +4|  = 7

Solution :

Let us write the given absolute value equation in the form

|x+a| = k

2|3x+4| = 7 -------> |3x+4| = 7/2

When we apply the method explained above for |3x + 4| = 7/2,

we get  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 = -5/2

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

Example 6 :

Solve the absolute value function 3|5x - 6|- 4  = 5

Solution :

Let us write the given absolute value equation in the form

|x+a| = k

3|5x-6|-4 = 5 -------> 3|5x-6| = 9 ------->|5x-6| = 3

When we apply the method explained above for |5x - 6| = 3,

we get  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

Now, let us look at a problem on "Solving absolute value equations" with quadratic polynomial in absolute sign.

Example 7 :

Solve the absolute value function |x² - 4x - 5| = 7

Solution :

When we apply the method explained above for |x²-4x-5| = 7,

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

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

Let us 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

Let us 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 given below.

Example 8 :

Solve the absolute value function 0.5|0.5x| - 0.5 = 2.5

Solution :

Let us write the given absolute value equation in the form

|x+a| = k

0.5|0.5x|-0.5=2.5 ------> 0.5|0.5x|=3 ------> |0.5x| = 6

When we apply the method explained above for |0.5x| = 6,

we get  0.5x  = 6   or    0.5x = -6

x = 12     or     x = -12

Hence the solution is x = -12, 12

Now let us look at some quiet different problems on "Solving absolute value equations"

Example 9 :

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

Solution :

Since x = -2 is a solution, we can plug x = -3 in the given absolute value equation |2x+k| = 3.

|2(-2)+k| = 3 -------|-4+k| = 3

Using the method explained above, we get

-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 :

Let us write the given absolute value equation in the form

|x+a| = k

|x-3|-k --------> |x-3| = k

Since x = 5 is a solution, we can plug x = 5 in the absolute value equation |x-3| = k

|-5-3| = k

|-8| = k

8 = k     (absolute value of any negative number is positive)

Hence the value of k = 8

After having gone through the step by step solutions for all the problems on "Solving absolute value equations",  we hope that the students would have understood how to do problems on "Solving absolute value equations".