How do you graph #y ≤ 3x + 5#?

1 Answer
May 24, 2017

graph{y<=3x+5 [-18.85, 17.2, -5.48, 12.54]}
Note that the area under the function is shaded.


We need to take a moment and not think of functions and x-y relations as just the relationship between x and y. Instead, think of it as a collection of points that satisfy a condition.

For example, let us suppose the line #y=x#
It can also be interpreted as a collection of all points whose #y# is equal to its #x#
In mathematics, we call this a set.
So, we can say that this is a set of all points who satisfy y=x, and we write it like this:
#{(x, y)|y=x}#

This is a very helpful intuition to have later on, for example with the circle equation:
graph{x^2+y^2=1 [-2.43, 2.435, -1.215, 1.217]}
Instead of thinking of this as a function #x^2+y^2=1#, think of it as a set of all points who satisfy that condition. That is to say that any point that lie on the circle (blue) must satisfy #x^2+y^2=1#.
#{(x, y)|x^2+y^2=1}#

Back to your inequality, think of it as a set of all points who satisfy #3y<=3x+5#. Think about it. It can't be on a single line or curve - instead its an entire area!

Start by drawing #y=(3x+5)/3#, then since its #y<=(3x+5)/3#, we shade in the underside of the line.