
Graphing Linear Inequalities: y > mx + b, etc First off, let mesay that graphing linear inequalites is much easier than your book makes it look. Here's how it works: Think about how you've done linear inequalites on the number line. For instance, they'd ask you to graph something like x > 2. How did you do it? You would draw your number line, find the "equals" part (in this case, x = 2), mark this point with the appropriate notation (an open dot or a parenthesis, indicating that the point x = 2 wasn't included in the solution), and then you'd shade everything to the right, because "greater than" meant "everything off to the right". The steps for graphing twovariable linear inequalities are very much the same.
Just as for numberline inequalities, my first step is to find the "equals" part. For twovariable linear inequalities, the "equals" part is the graph of the straight line; in this case, that means the "equals" part is the line y = 2x + 3:
Now we're at the point
where your book probably gets complicated, with talk of "test points"
and such. But when you did those onevariable inequalities (like x < 3), you didn't
bother with "test points"; you just shaded one side or the other.
We can do the same here. Ignore the "test point" stuff, and
look at the original inequality: y < 2x + 3. I've already graphed the "or equal to" part (it's just the line); 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 it in: And that's all there is to it: the side I shaded is the "solution region" they want. This technique worked because we had y alone on one side of the inequality. Just as with plain old lines, you always want to "solve" the inequality for y on one side.
First, I'll solve for y: 2x – 3y < 6 [Note 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!] Copyright © Elizabeth Stapel 20002011 All Rights Reserved Now I need to find the "equals" part, which is the line y = (^{ 2}/_{3} )x – 2. It looks like this: But this exercise is what is called a "strict" inequality. That is, it isn't an "or equals to" inequality; it's only "y greater than". When I had strict inequalities on the number line (such as x < 3), I denote 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 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. Since this is a "y greater than" inequality, I want to shade above the line, so my solution looks like this:
If you need to graph a set of two or more linear inequalities at once, view the lesson on systems of linear inequalities.


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