How to Graph 1- and 2-Variable Inequalities

All right, so you know that an inequality is an equation with a greater than (>) or less than ( 10 is different than 10 > x .

Therefore, there is one Golden Rule about inequalities that make them different than equations.

Anytime you multiply or divide both sides of the inequality by a negative number, you must flip the inequality symbol around.

If we would have used this rule the first time, our answer would have instead said x is greater than or equal to 6.

Just to make sure that works out, let’s check it real fast.

So, 10 is bigger than 6, so if we substitute 10 into our original equation, 5 + 10 = -5 and -5 is less than or equal to -1, so we’re in good shape.

Graphing our solution now simply requires us to draw a number line, work our way over to 6, put a circle there, fill it in because it’s ‘or equal to’, and then draw an arrow to the right, indicating anything bigger than 6 is good to go.

The last example for this video will be a 2-variable inequality.

I bet you can guess what is going to be different about this one.

Instead of just x s, we’ll have x s and y s!

Maybe something like, graph y < 2 x - 5.

Just like in the first two examples, we need to find the boundary where our bad answers end and our good answers begin.

But because we now have two variables, our graph will have two axes.

This means that the boundary is no longer a point like before; it’s a line.

And this boundary right in between where y goes from being less than to greater than is where y = 2 x - 5.

I begin at the y -intercept, -5, and I use the slope to go up 2 and over 1 and I find my line right where y = 2 x - 5.

So, what we have here is the boundary where y = 2 x - 5, but we want to know where it is less.

The best way to determine which side of the boundary is the less than side is by picking a random point, plugging it in, and seeing if it works in our inequality.

The origin, (0, 0), is often a good one to try because plugging in 0s make a lot of the math easy, so going back to our original inequality and substituting in 0s gives us 0 < - 5, which is most certainly not true.

That means that (0, 0) does not live in the ‘less than land,’ which means it lives in the ‘greater than land;’ therefore, the remaining section of the graph must be what we’re looking for, and y < 2 x - 5 is the area underneath the boundary.

But also like the 1-variable problem, we need to indicate whether or not the inequality had an ‘or equal to’ on it.

This example did not, so we need to make it clear that the boundary line where it’s equal to is not part of the answer.

We’d like to just erase it, but then we wouldn’t know where the ‘less than land’ ends and the ‘greater than land’ begins, so instead we make the equal line a dotted line.

If the problem would have had the ‘or equal to’ we would have simply left the line as it was and been done.

To review:

1-variable inequalities can be solved just like 1-variable equations, with the only difference being that you must switch the inequality symbol around when multiplying or diving both sides by a negative number.

1-variable inequalities are graphed on a number line, with a circle representing the boundary and an arrow indicating which direction the solutions are in.

2-variable inequalities are graphed on an x-y coordinate plane, with a line representing the boundary and shading indicating which side the solutions are on.

Lastly, ‘strictly less than’ or ‘strictly greater than’ means an empty circle or a dotted line to represent the boundary in your graph, while having an ‘or equal to’ means a filled-in circle or a solid line as the boundary.