# How do you graph the inequality 3-x>0 and y+x< -6?

Jan 22, 2018

See below.

#### Explanation:

First we write the inequalities as linear equations, and the graph these. Remember to use a dashed line, as these are not equal to inequalities, so the line itself will not be an included region.

$3 - x > 0$

$3 - x = 0$ , $x = 3 \textcolor{w h i t e}{88888888888}$ First equation.

$y + x < - 6$

$y + x = - 6$ , $y = - x - 6 \textcolor{w h i t e}{88}$ Second equation.

We now graph these:

With graph plotted, we can see that there are four possible regions.

A , B , C and D

The required region has to satisfy both inequalities, so we test a set of coordinates in each region. We can save some work by realising that if a coordinate in a region fails for the first inequality we test, then the region can't be the required region and it is not necessary to test the other inequality as well.

Region A

Coordinates: $\left(- 2 , 2\right)$

$\textcolor{b l u e}{3 - x > 0}$

$3 - \left(- 2\right) > 0 \textcolor{w h i t e}{88}$ , $5 > 2 \textcolor{w h i t e}{88888888}$ TRUE

$\textcolor{b l u e}{y + x < - 6}$

$2 + \left(- 2\right) < - 6 \textcolor{w h i t e}{88}$ ,$0 < - 6 \textcolor{w h i t e}{888}$ FALSE

Region A is not the required region.

Region B

Coordinates: $\left(2 , 2\right)$

$\textcolor{b l u e}{3 - x > 0}$

$3 - \left(2\right) > 0 \textcolor{w h i t e}{88}$ , $1 > 0 \textcolor{w h i t e}{888888888}$ TRUE

$\textcolor{b l u e}{y + x < - 6}$

$2 + 2 < - 6 \textcolor{w h i t e}{88}$ , $4 < - 6 \textcolor{w h i t e}{888} \textcolor{w h i t e}{88}$ FALSE

Region B is not the required region.

Region C

Coordinates: $\left(- 2 , - 6\right)$

$\textcolor{b l u e}{3 - x > 0}$

$3 - \left(- 2\right) > 0 \textcolor{w h i t e}{88}$ , $5 > 0 \textcolor{w h i t e}{888888888888888}$ TRUE

$\textcolor{b l u e}{y + x < - 6}$

$\left(- 6\right) + \left(- 2\right) < - 6 \textcolor{w h i t e}{88}$ , $- 8 < - 6 \textcolor{w h i t e}{8888}$ TRUE

Region C is a required region.

We do not need to test region D. As can be seen from the graph above if C is an included region and is to the left of the line $x = 3$, then to the right of this line is an excluded region.