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# Absolute Value Inequalities

Absolute value of a number shows its distance from the  origin on the number line. The negative value will be on the left of the origin and the positive ones on the right. Any inequality of an absolute term will fall either between positive and negative value of a number of outside it. Absolute value inequalities defines a range of possible values of the variable on a real line. There are two forms of absolute value inequalities. One with less than, |a|< b, and the other with greater than, |a|> b. The absolute value of a number measures its distance to the origin and absolute value inequality gives range of possible values of the variable on a real line.

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## Solving Absolute Value Inequalities

In order to solve an absolute value inequality we need two concepts:
i) Solving an absolute value problem ii) Solving an inequality problem.
Generally, we will solve inequalities one of two ways depending on the type of problem.

## Graphing Absolute Value Inequalities

To graph an absolute value inequality, we need to find absolute value inequality function first.

If it is given for a number k,
|x| < k then the function will be written as -k < x < k
|x| > k then the function will be x < -k and x > k

Steps For Graphing Absolute Value Inequalities:

Step 1: Find absolute value inequalities in terms of positive and negative values.

Step 2: Find the range in which inequality will hold true.

Step 3: Place the obtained range on the number line.

### Solved Examples

Question 1: Find all real number solutions to |x + 2| $\leq$ 5 and sketch the solution on number line.

Solution:

Given inequality |x + 2| $\leq$ 5

Step 1: Change the absolute value inequality into a compound inequality

- 5 $\leq$ x + 2 $\leq$ 5

Step 2: Subtract 2 from all three sides to get

- 5 -2  $\leq$ x + 2 - 2  $\leq$ 5 - 2

= - 7  $\leq$ x  $\leq$ 3

Step 3: Place this on a number line

Question 2: Find all real number solutions to |x + 10| > 7 and sketch the solution on number line.
Solution:

Given inequality |x + 10| > 7

which is similar to the |x| > a, a > 0

So, solution for |x| > a are x > a or x < - a

Step 1:

x + 10 > 7  or  x + 10 < - 7

Step 2:   Subtract 10 from above

x + 10 - 10 > 7 - 10  or  x + 10 - 10 < - 7 - 10

x > - 3  or  x < - 17

Step 3: Place this on a number line

Question 3:
Find all real number solutions to 16 $\geq$ 2 |6 - x| and sketch the solution on number line.

Solution:

Given inequality 16 $\geq$ 2 |6 - x|

or $\frac{16}{2}$ $\geq$ |6 - x|

or 8 $\geq$ |6 - x|

Step 1: Change the absolute value inequality into a compound inequality

- 8 $\leq$ 6 - x $\leq$ 8

Step 2:   Subtract 6 from above

- 8 - 6 $\leq$ 6 - x - 6 $\leq$ 8 - 6

-14 $\leq$ - x $\leq$ 2

Step 3:
Operate with negative sign, which change the inequality

- 2 $\leq$ x $\leq$ 14

Step 4: Place this on a number line

## Problems

Absolute value inequalities are solved as similar linear inequalities. Lets study with the help of solved problems.

### Solved Examples

Question 1: Solve |2x + 10| $\geq$ 14.

Solution:

Given |2x + 10| $\geq$ 14

This inequality is similar to the |x| $\geq$ a, a > 0

So, solution for |x| $\geq$ a are x $\geq$ a or x  $\leq$ -a

Step 1:

2x + 10 $\geq$ 14 or 2x + 10 $\leq$ - 14

Step 2: Subtracting 10 from all

2x + 10 - 10  $\geq$ 14 - 10  or  2x + 10 - 10  $\leq$ - 14 -10

2x $\geq$ 4  or  2x $\leq$ - 24

Step 3:
Dividing by 2, we have

x $\geq$ 2  or  x $\leq$ - 12

Question 2:
If x is an integer, what is the solution to

|x + 4|  > 5

Solution:

Given |x + 4|  > 5

This inequality is similar to the |x| > a, a > 0

So, solution for |x| > a are x > a or x  < -a

Step 1:

x + 4 > 5  or x + 4 < - 5

Step 2:  Subtract 4 from above

x + 4 - 4 > 5 - 4  or  x + 4 - 4 < - 5 - 4

x > 1  or  x < - 9

Question 3:
Find the solution for

|x - 1| <  3, If x is an integer.

Solution:

Given |x - 1| <  3

This inequality is similar to the |x| < a, a > 0

So, solution for |x| < a are x < a and x  > -a

Step 1:
x - 1 > 3 and x - 1 < -3

or -3 < x - 1 < 3

Step 2:

-3 + 1 < x - 1 + 1 < 3 + 1

-2 < x < 4

or x $\in$ (-2, 4).

Question 4: Find the solution for

|2x - 3| $\leq$ 11, If x is an integer.

Solution:

Given |2x - 3| $\leq$ 11

This inequality is similar to the |x| $\leq$ a, a > 0

So, solution for |x| $\leq$ a are x $\leq$ a and x  $\geq$ -a

Step 1:
2x - 3 $\leq$ 11  or  2x - 3 $\geq$ -11

or -11 $\leq$ 2x - 3 $\leq$ 11

Step 2: By adding 3, we have

-11 + 3 $\leq$ 2x - 3 + 3 $\leq$ 11 + 3

- 8 $\leq$ 2x $\leq$ 14

Step 3: Dividing by 2,

-4 $\leq$ x $\leq$ 7

or x $\in$ [-4, 7].

## Practice Problems

Problem 1: Graph the given inequality on a number line.
|3x - 7| < -3

Problem 2: Find the function for the given inequality and graph it.
|4x - 5| > x + 2

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