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

1 Answer
Aug 11, 2017

Graph the equation #y=x-6# then shade the side of the graph for which #y<=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:
enter image source here

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

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

Is #(x,y)=(0,0)# a valid solution point for #y<=x-6#?
That is, is #0 <= -6#?
No.
Therefore, the side of the line of equality containing #(0,0)# 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<=x-6#
enter image source here