# How do you graph and solve 2x-1-abs[9-3x]<abs[3x]?

Mar 21, 2017

See below.

#### Explanation:

$2 x - 1 - 3 \left\mid x - 3 \right\mid < 3 \left\mid x \right\mid$ now supposing $x \ne 0$

$2 \frac{x}{\left\mid x \right\mid} - \frac{1}{\left\mid x \right\mid} - 3 \frac{\left\mid x - 3 \right\mid}{\left\mid x \right\mid} < 3$ but

$\frac{x}{\left\mid x \right\mid} = \pm 1$ so

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

$\left\mid x \right\mid \max \left(- 5 , - 2\right) < 1 + \left\mid x - 3 \right\mid$ or

$- 2 \left\mid x \right\mid < 1 + \left\mid x - 3 \right\mid$ or

$2 \left\mid x \right\mid + \left\mid x - 3 \right\mid + 1 > 0$

So it is true for all $x$