# How do you solve 5< 3- 2x/3?

Jul 16, 2015

$x < - 12 \mathmr{and} x < - 3$
$\textcolor{w h i t e}{\text{XXXX}}$depending upon interpretation of $2 \frac{x}{3}$

#### Explanation:

Note I have interpreted $2 \frac{x}{3}$ as a mixed fraction; if it were intended to be a multiplication, see Solution 2 below.

Solution 1 ($2 \frac{x}{3}$ as a mixed fraction)
If $5 < 3 - 2 \frac{x}{3}$$\textcolor{w h i t e}{\text{XXXX}}$$\rightarrow \left(5 < 3 - \frac{6 + x}{3}\right)$
We can clear the fraction by multiplying everything by 3 (without effecting the validity of the inequality orientation)
$\textcolor{w h i t e}{\text{XXXX}}$$15 < 9 - 6 - x$

Subtract 3 from both sides (again, no effect on the validity of the inequality orientation)
$\textcolor{w h i t e}{\text{XXXX}}$$12 < - x$

Multiply both sides by $\left(- 1\right)$ (This will reverse the inequality orientation. Note that as an alternative we could have added $\left(x - 12\right)$ to both sides without effecting the orientation and achieve the same result)
$\textcolor{w h i t e}{\text{XXXX}}$$x < - 12$

Solution 2 ($2 \frac{x}{3}$ as implied multiplication)
If $5 < 3 - 2 \frac{x}{3}$$\textcolor{w h i t e}{\text{XXXX}}$$\rightarrow \left(5 < 3 - \frac{2}{3} x\right)$

Subtract 3 from both sides
$\textcolor{w h i t e}{\text{XXXX}}$$2 < - \frac{2}{3} x$

Multiply both sides by $\left(- \frac{3}{2}\right)$ (forcing a reversal of the inequality orientation
$\textcolor{w h i t e}{\text{XXXX}}$$- 3 > x$
or
$\textcolor{w h i t e}{\text{XXXX}}$x < -3#