Absolute Value Inequalities

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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

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]

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