# Absolute Value Inequalities

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Absolute value will always give a positive value that means whether a value is positive or negative but when

you take a positive value of that particular number then it will gives a positive value only.

Example of absolute value inequalities: -

Question 1: - Solve the inequalities | 3 x + 5 | < 9.

Solution: - If 3 x + 5 > 0      then | 3 x + 5 | < 9 gives,

3 x + 5 < 9      or, 3 x < 4       or, x < 4 / 3     … (1)

Again, if 3 x + 5 < 0,             then | 3 x + 5 | < 9 gives

- (3 x + 5) < 9 or, - 3x < 14   or, x > - 14 / 3            … (2)

Therefore from (1) and (2) we get,  -14 / 3 < x < 4 / 3, which is required solution of the given inequalities.

Question 2: - If x is an integer and | x + 1 | < 3, find x.

Solution: - If | x + 1 | < 3 then we can write

-3< (x + 1) < 3

-3 – 1 < x < 3 – 1

- 4 < x < 2

Question 3: - If x is a real number and | 2 x – 7 | < 1, find x.

Solution: - If | 2 x – 7 | < 1 then

-1 < (2 x – 7) < 1

-1 + 7 < 2 x < 1 + 7

6 < 2 x < 8

3 < x < 4