# 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:

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

### Answer Key

1) |2x-1|<=3

-3 ≤ 2*x* – 1 ≤ 3

-3 ≤ 2*x* – 1 and 2*x* -1 ≤ 3

Adding 1 to both sides of an inequality, we get

-2 ≤ 2*x* and 2*x* ≤ 4

*x *≥ -1 and *x* ≤ 2

Interval Notation: [-1,2]

2) 2|3x+2|>=16

Dividing by 2 on both sides of an inequality, we get

|3*x* + 2| ≥ 8

-8 ≥ 3*x* + 2 ≥ 8

-8 ≥ 3*x *+ 2 and 3*x *+ 2 ≥ 8

-10 ≥ 3*x* and 3*x* ≥ 6

*x* ≤-(10/3) and *x *≥ 2

Interval notation:[(-∞,(-10/3))U(2,∞)]

3) |*x *+ 2|+4<6

|*x *+ 2| < 2

-2 < *x *+ 2 < 2

-2 < *x* + 2 and *x* + 2 < 2

*x* > -4 and *x* < 0

Interval notation:(-4,0)

4) |2*x* – 3|-1>8

|2*x* – 3| > 9

-9 > 2*x* – 3 and 2*x* -3 > 9

*x* < -3 and *x* > 6

Interval notation: (-∞,-3) U (6,∞)

5) |5*y* + 2| +1<=13

|5*y* + 2| ≤ 12

-12 ≤ 5*y* + 2 ≤ 12

-12 ≤ 5*y* + 2 and 5*y* + 2 ≤ 12

y>=(-14/5) and y<=2

Interval notation:[(-14/5),2]