How do you solve the inequality abs(3-x)>=x-5 and write your answer in interval notation?

Mar 21, 2017

See below.
The inequality is true $\forall x \in \mathbb{R}$

Explanation:

We have $\left\mid x - 3 \right\mid \ge x - 3 - 2$ Now supposing $x \ne 3$ we can do

$\frac{\left\mid x - 3 \right\mid}{\left\mid x - 3 \right\mid} \ge \frac{x - 3}{\left\mid x - 3 \right\mid} - \frac{2}{\left\mid x - 3 \right\mid}$ or

$1 \ge \pm 1 - \frac{2}{\left\mid x - 3 \right\mid}$

or

$1 + \frac{2}{\left\mid x - 3 \right\mid} \ge \pm 1$

or

$1 \le \frac{2}{\left\mid x - 3 \right\mid} + 1$

or

$\left\mid x - 3 \right\mid \le 2 + \left\mid x - 3 \right\mid$

or

$2 \ge 0$ which is true for all $x$