Question

How do you graph `y ≤ 3x + 5`?

Answer 1

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

Explanation:

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.