# How do you solve the inequality 9 - y >3 - 2y  and  -1 - 2y > -5?

May 8, 2017

$y \in \left(- 6 , 2\right)$

#### Explanation:

Remember:
You can add or subtract any amount to both sides of an inequality
and
multiply or divide both sides of an inequality by any positive amount
without changing the validity or orientation of the inequality.

$\textcolor{g r e e n}{\text{============================================}}$

Part 1: color(white)("xxx")color(black)(9-y > 3-2y

$\textcolor{w h i t e}{\text{XXX}} 9 - y \textcolor{m a \ge n t a}{+ 2 y} > 3 - 2 y \textcolor{m a \ge n t a}{+ 2 y}$

$\textcolor{w h i t e}{\text{XXX}} 9 + y > 3$

$\textcolor{w h i t e}{\text{XXX}} 9 + y \textcolor{m a \ge n t a}{- 9} > 3 \textcolor{m a \ge n t a}{- 9}$

$\textcolor{w h i t e}{\text{XXX}} y > - 6$

$\textcolor{g r e e n}{\text{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}}$

Part 2: $\textcolor{w h i t e}{\text{xxx}} \textcolor{b l a c k}{- 1 - 2 y > - 5}$

$\textcolor{w h i t e}{\text{XXX}} - 1 - 2 y \textcolor{m a \ge n t a}{+ 2 y} > - 5 \textcolor{m a \ge n t a}{+ 2 y}$

$\textcolor{w h i t e}{\text{XXX}} - 1 \succ 5 + 2 y$

$\textcolor{w h i t e}{\text{XXX}} - 1 \textcolor{m a \ge n t a}{+ 5} > - 5 + 2 y \textcolor{m a \ge n t a}{+ 5}$

$\textcolor{w h i t e}{\text{XXX}} 4 > 2 y$

$\textcolor{w h i t e}{\text{XXX}} \frac{4}{\textcolor{m a \ge n t a}{2}} > \frac{2 y}{\textcolor{m a \ge n t a}{2}}$

$\textcolor{w h i t e}{\text{XXX}} 2 > y$

$\textcolor{g r e e n}{\text{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}}$

Part 3: $\textcolor{w h i t e}{\text{xxx")[9-y > 3-2y]color(white)("XX}}$and $\textcolor{w h i t e}{\text{XX}} \left[- 1 - 2 y > - 5\right]$

$\textcolor{w h i t e}{\text{XXX")2 > ycolor(white)("XX}}$and$\textcolor{w h i t e}{\text{XX}} y > - 6$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow 2 > y > - 6$

$\textcolor{w h i t e}{\text{XXX}} \mathmr{and} - 6 < y < 2$

$\textcolor{w h i t e}{\text{XXX}} \mathmr{and}$ (in interval notation) $y \in \left(- 6 , 2\right)$