# How do you graph the inequality y<=x-6?

Aug 11, 2017

Graph the equation $y = x - 6$ then shade the side of the graph for which $y \le x - 6$ is true.

#### Explanation:

For $y = x - 6$ we select some sample points (2 would be enough, but I will evaluate with 3 for safety):
color(white)("XXX"){:(ul(x),color(white)("xxxx"),ul(y=x-6)), (6,,0), (3,,-3), (0,,-6) :}
Now graph these points and draw a straight line through them:

Next we need to shade (select) the side of this line for which $y$ is less than or equal to $\left(x - 6\right)$

Consider an arbitrary point not on the line of equality.
For a case like this I often like to use $\left(x , y\right) = \left(0 , 0\right)$

Is $\left(x , y\right) = \left(0 , 0\right)$ a valid solution point for $y \le x - 6$?
That is, is $0 \le - 6$?
No.
Therefore, the side of the line of equality containing $\left(0 , 0\right)$ must be the side that should not be included.

Shade the other side (remembering to leave, the equality line solid since it represents valid solution points for $y \le x - 6$