Question

How do you solve and write the following in interval notation: `| 2x - 3 |<4`?

Answer 1

The absolute value of a number depends on whether the number is positive, zero or negative. Then you have to take the given inequality into account.

Explanation:

`| a | = a`, if `a > 0`
`| a | = -a`, if `a < 0`
`| 0 | = 0`

There are then 3 cases to have a look at:
`2x - 3 > 0`, thus `2x > 3`, then `x>3/2`

In this case `|2x-3| = 2x-3`
and the inequality is, `2x - 3 < 4`, so
`2x < 4 + 3`, or `2x < 7`, that is
`x < 7/2`.
Thus, at the same time , `3/2 < x` and `x<7/2`, thus the elements `x` between `3/2` and `7/2` satisfy the inequality.

Similarly, if `2x - 3 < 0`, thus `2x < 3`, then `x<3/2`
In this case `|2x-3| = -2x+3`
and the inequality is, `-2x + 3 < 4`, so
`-2x < 4 - 3`, or `-2x < 1`, that is
`-1< 2x` and thus `-1/2 < x`
Then, at the same time , `-1/2 < x` and `x<3/2`, thus the elements `x` between `-1/2` and `3/2` satisfy the inequality.

Finally, when `2x-3=0`, `x=3/2`, and plugging the value `3/2` into the inequality we get `0 < 4`, which is true, and then `x=3/2` also satisfies the inequality.

Summarising, ALL the elements `x` between `-1/2` and `7/2` satisfy the inequality; that is the open interval (`-1/2`, `7/2`)