Question

Let f(x) = 13 - |2x+ 1|. Find all x for which f(x) less than or equal to 14

Answer 1

`x \leqslant -1\: or \: x \geqslant - 1`

Explanation:

The given function is

`f(x) = 13 - |2x + 1|`

We want to find all values of x for which:

f(x) is greater than or equal to 14.

This implies;

`13 - |2x + 1| \geqslant 14`

Subtract 13 from both sides;

`- |2x + 1| \geqslant 14 - 13`

`|2x + 1| \geqslant - 1`

By definition of the absolute value function,

`- (2x + 1) \geqslant - 1 \: or \: (2x + 1) \geqslant - 1`

Divide through the first inequality and and reverse the inequality sign:

`2x + 1 \leqslant -1 \: or \: 2x + 1 \geqslant - 1`

`2x \leqslant -1 - 1 \: or \: 2x \geqslant - 1 - 1`

`2x \leqslant -2 \: or \: 2x \geqslant - 2`

`x \leqslant -1\: or \: x \geqslant - 1`