# How do you graph y ≤ 3x + 5?

May 24, 2017

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:
$\left\{\left(x , y\right) | y = x\right\}$

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 ${x}^{2} + {y}^{2} = 1$.
$\left\{\left(x , y\right) | {x}^{2} + {y}^{2} = 1\right\}$

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

Start by drawing $y = \frac{3 x + 5}{3}$, then since its $y \le \frac{3 x + 5}{3}$, we shade in the underside of the line.