Question

How do you solve the absolute value inequality `abs(2x - 3)< 5`?

Answer 1

The result is `-1 < x < 4`.

The explanation is the following:

In order to be able to supress the absolute value (which is always disturbing), you can apply the rule: `|z| < k, k in RR => -k < z < k`.
By doing this you have that `|2x-3| < 5 => - 5 < 2x-3 < 5`, which are two inequalities put together. You have to solve them separately:
1st) `- 5 < 2x-3 => - 2 < 2x => - 1 < x`
2nd) `2x-3 < 5 => 2x < 8 => x < 4`
And, finally, putting both results together (which is always more elegant), you obtain the final result which is `- 1 < x < 4`.

Answer 2

The result is `-1 < x < 4`.

The explanation is the following:

In order to be able to supress the absolute value (which is always disturbing), you can apply the rule: `|z| < k, k in RR => -k < z < k`.
By doing this you have that `|2x-3| < 5 => - 5 < 2x-3 < 5`, which are two inequalities put together. You have to solve them separately:
1st) `- 5 < 2x-3 => - 2 < 2x => - 1 < x`
2nd) `2x-3 < 5 => 2x < 8 => x < 4`
And, finally, putting both results together (which is always more elegant), you obtain the final result which is `- 1 < x < 4`.