HOW TO SOLVE INEQUALITIES WITH MODULUS

Solving Inequalities with Modulus - Concept

If the given question is in any of the following forms, we have to follow the given methods to solve for x.

Example 1 :

Solve the absolute value inequality given below

|x - 9| < 2

and express the solution in interval notation.

Solution :

-2 < x - 9 < 2

Add 9 throughout the equation

-2 + 9 < x - 9 + 9 < 2 + 9

7 < x < 11

Hence the solution set of the above absolute inequality is (7, 11).

Example 2 :

Solve the absolute value inequality given below

|2/ (x - 4)|  > 1 , x ≠ 4 

and express the solution in interval notation.

Solution :

From the given inequality, we have that 2 > (x - 4)

-2 < x - 4 < 2

Add 4 throughout the inequality

-2 + 4 < x - 4 + 4 < 2 + 4

2 < x  < 6

We cannot express the solution as (2, 6). Because in the middle of 2 and 6, we have the value 4.

So, we have to split it into two intervals.

(2, 4) U (4, 6)

Example 3 :

Solve the absolute value inequality given below

|3 - (3x/4)| ≤  1/4

and express the solution in interval notation.

Solution :

(-1/4) ≤ 3 - (3x/4)  ≤ (1/4)

(-1/4) ≤ (12 - 3x)/4  ≤ (1/4)

Multiply by 4 throughout the equation

-1  ≤ (12 - 3x) ≤ 1

Subtract 12 throughout the equation

-1 - 12 ≤ 12 - 3x - 12 ≤ 1 -12

-13 ≤ - 3x  ≤ -11

Divided by (-3) throughout the equation

-13/(-3) ≤ - 3x  ≤ -11

13/3 ≤ x ≤ 11/3

11/3 ≤ x ≤ 13/3

Hence the solution set of the above absolute inequality is [11/3, 13/3].

Example 4 :

Solve the absolute value inequality given below

|6x + 10| ≥  3

and express the solution in interval notation.

Solution :

6x + 10 ≤ -3  and 6x + 10  ≥  3

Hence the solution set of the above absolute inequality is (-∞, -13/6] U [-7/6, ∞). 

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