Question

How do you solve compound inequalities `-2x + 1 > 3` and `-2x + 1 < 9`?

Answer 1

Using the shorthand notation `3 < -2x + 1 < 9`, first subtract 1 from all parts to give: `2 < -2x < 8`.

Next, divide all parts by -2 and reverse the inequalities (since -2 is negative), to give:
`-1 > x > -4`, i.e. `-4 < x < -1`.

The truth of a multi-part inequality is preserved by the following operations:
(1) Add or subtract a given number from all parts.
(2) Multiply or divide all parts by a positive number.
(3) Multiply or divide all parts by a negative number and reverse the inequalities.