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 ≤ 2x – 1 ≤ 3
-3 ≤ 2x – 1 and 2x -1 ≤ 3
Adding 1 to both sides of an inequality, we get
-2 ≤ 2x and 2x ≤ 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
|3x + 2| ≥ 8
-8 ≥ 3x + 2 ≥ 8
-8 ≥ 3x + 2 and 3x + 2 ≥ 8
-10 ≥ 3x and 3x ≥ 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) |2x – 3|-1>8
|2x – 3| > 9
-9 > 2x – 3 and 2x -3 > 9
x < -3 and x > 6
Interval notation: (-∞,-3) U (6,∞)
5) |5y + 2| +1<=13
|5y + 2| ≤ 12
-12 ≤ 5y + 2 ≤ 12
-12 ≤ 5y + 2 and 5y + 2 ≤ 12
y>=(-14/5) and y<=2
Interval notation:[(-14/5),2]