Absolute Value Inequality

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An inequality | x – a| < b is known as absolute value inequality where | x – a | is known as the absolute value

which always gives a positive value.

 
Example of absolute value inequality: -
 
·         | x – 1 | < 0

·         | x + 1 | + | x | < 2 x

·         | x + 2 | < 2

·         | x + 3 | > 2

 
Question 1: - Solve | x – 3 | > 6

 
Solution: - | x – 3 | > 6

Case 1: -        +(x – 3) > 6

                        x – 3 > 6

                        x> 6 + 3

                        x> 9

Case 2: -        - (x – 3) > 6

                         (x – 3) < - 6

                          x< 3 – 6

                          x< - 3

 
Question 2: - Solve | x + 1 | < 2 x

 
Solution: - If | x - 1 | < 2 x, then

Case 1: -        + (x - 1) < 2x

                        X – 2 x < 1

                       -       x< 1

                               x > -1

Case 2: -        - (x - 1) < 2x

                         (x - 1) > - 2x

                          x+ 2x >1

                           3 x >1

                              x> 1/3

 
Question 3: - If | 2 x | < 2 find x.

 
Solution: -     If | 2 x | < 2 then

Case 1: -        + (2x) < 2        so, x < 1

Case 2: -        - (2x) < 2         or, 2 x > - 2     so, x > -1

Therefore       -1 < x < 1

 

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