# How do you graph -y+3x<=6?

Apr 1, 2015

There are a couple possible approaches. Here's one.

I finding leading negative signs hard to read, so I would begin by rewriting the inequality as: $3 x - y \le 6$

Start by graphing the equation: $3 x - y = 6$

For this equation, it is straightforward to find the intercepts, so that's now I would graph this one.
(If you prefer to put it in slope-intercept form first, do that.)

$\left(0 , - 6\right)$ and $\left(2 , 0\right)$ are the intercepts so draw the line through those two points. So you get this:

graph{3x-y = 6 [-10, 10, -5, 5]}

The line $3 x - y = 6$ cuts the plane into two regions. In one region, the value of $3 x - y$ is $< 6$, in the other it is $> 6$. Our job now is to figure out which side is which so we can stay on the "less than 6" side.

I see that the point $\left(0 , 0\right)$ (the origin) is not on the graph of the equation, so I'll just check to see if that side is the $< 6$ or $> 6$ side.
$3 \left(0\right) - \left(0\right) = 0 - 0 = 0$ which is less than $6$. So the region above the line must be the $< 6$ side of the line.

The inequality we're looking at wants the $\le 6$ side, so we shade that side. (If you wanted to double check, you could pick a point above the line. Say $\left(5 , 0\right)$ or (10, 0) and make sure that 3x - y < 6#

Your graph should look like this:

graph{3x-y<=6 [-10, 10, -5, 5]}