First off, let me say that graphing two-variable linear inequalites in the *x*,*y*-plane is much easier than your textbook makes it seem.

Think about how you've solved one-variable linear inequalites on the number line. Suppose they'd asked you to graph something like *x* > 2. How did you do that?

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You found your solution by drawing your number line and finding the "equals" part (in this example, *x* = 2). Then you marked this point with the appropriate notation; namely, an open dot or a parenthesis, indicating that the point *x* = 2 wasn't included in the solution. Then you'd shade everything to the right, because "greater than" meant "everything off to the right".

The steps for graphing two-variable linear inequalities are very much the same as for graphing the one-variable case.

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- Solve the inequality to get
*y*alone on one side. - Replace the inequality symbol with an "equals" sign, creating a straight-line equation.
- Graph the equation.
- If the original inequality was an "or equal to" one, then draw a solid line for the graph.
- If, on the other hand, the original inequality was a "strict" inequality, then draw a dashed line for the graph.

- Shade one side of the straight line.
- If the solved inequality was "
*y*greater than", then shade above the line. - If the solved inequality was "
*y*less than", then shade below the line.

- If the solved inequality was "

- Graph the solution to
*y*≤ 2*x*+ 3.

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* = 2*x* + 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:

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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* ≤ 2*x* + 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:

The plane, containing the line and a shaded side, is the "solution region" they're wanting. That's all there is to it!

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This technique (of looking at the original inequality to see which side to shade) worked because we had *y* alone on one side of the inequality. Just as was the case back when you were graphing plain old straight lines, you always want to solve for *y* to be alone on one side. Doing so makes everything else simpler and, for inequalities like this, saves the annoyance of doing test points.

- Graph the solution to 2
*x*− 3*y*< 6.

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

2*x* − 3*y* < 6

−3*y* < −2*x* + 6

[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.

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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:

Related page: If you need to graph a set of two or more linear inequalities at once, view the lesson on systems of linear inequalities.

URL: https://www.purplemath.com/modules/ineqgrph.htm

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