Absolute Value Inequalities

An inequality that has an absolute value sign, with a variable inside is called an absolute value inequality.

The type of inequality signs shown in the below cases, explains how to set up a compound inequality.

Case I:|x-a|<=b

Then, we can set compound inequality as below:

-b<=x-a<=b

Case II:   |x-a|>=b

Then, we can set compound inequality as below:

x-a <= -b  Or  x-a >= b

Example 1:     6<=3x<=15

6<=3x  And 3x<=15

Dividing both sides by 3, we get

2<=x   And x<=5

2<=x<=5 

Interval notation =>[2,5]

Example 2:   2(y-1) <6  or  2(y-1) >10

These inequalities are connected with < or > symbols, so the solution will be the union of solutions of these two inequalities.

Dividing both sides by 2, we get

(Y-1)< 3 or (y-1)> 5

Adding 1 to both sides of an inequality gives,

y<4   Or  y>6

Interval notation =>(-∞,4) U (6,∞)

Note: Absolute value term is always non-negative.

Check Point

 Solve the given absolute value inequalities and write the answer in interval notation:

1. |2x-1|<=3
2. 2|3x+2|>=16
3. |x+2|+4<6
4. |2x-3|-1>8
5. |5y+2|+1<=13