Graphing Two-Variable Linear Inequalities

Just as for one-variable linear number-line inequalities, my first step for this two-variable linear x,y-plane inequality is to find the "equals" part of the inequality. For two-variable linear inequalities, the "equals" part is the graph of the straight line; in this case, that straight line is y = 2x + 3.

And, because this particular inequality is an "or equal to" inequality, this tells me that the straight line is included in the solution; in particular, I should draw my line as a solid line, rather than as a dashed line:

Okay; the line has given me the boundary between the solution region (being the portion of the plane that solves the inequality) and the not-a-solution region.

Now I'm at the point where the textbook tends to get complicated, with talk of "test points" and such. But back when I did one-variable inequalities (like x < 3), I didn't bother with "test points"; I just shaded one side or the other. I can do the same here.

I'll ignore the "test point" stuff, and instead look at the original inequality, y ≤ 2x + 3.

I've already graphed the "or equal to" part — it's just the line, which is the border of the solution region. Now I'm ready to do the "y less than" part. In other words, this is where I need to shade one side of the line or the other.

Now think about it: If I need y less than the line, do I want above the line, or below?

Naturally, I want below the line. So I shade in that portion of the plane:

First, I'll solve the inequality to get y alone on one side:

[Notice how I the flipped inequality sign in the last line. I mustn't forget to flip the inequality if I multiply or divide through by a negative.]

Now I need to find the "equals" part, which is the line y = (²/₃)x − 2. That line looks like this:

Okay; the "equals" part has given me the line, which is the boundary between the solution region and the not-a-solution region.

But this exercise is what is called a "strict" inequality. That is, it isn't an "or equal to" inequality; it's only "y greater than". So this affects the "equals" line.

When I had strict inequalities on the number line (such as x < 3), I denoted this by using a parenthesis (instead of a square bracket) or an open [unfilled] dot (instead of a closed [filled] dot).

In the case of these two-variable linear inequalities, the notation for a strict inequality is a dashed line. So the border of my solution region actually looks like this:

By using a dashed line, I still know where the border is, but I also know that the border isn't included in the solution. And, since this is a "y greater than" inequality, I know that I want to shade above the line, so my solution looks like this: