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 thenumber 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 theappropriate 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 "everythingoff to the right". The steps for graphing two-variable linear inequalities are very much the same.
Graph the solution to y < 2x + 3.
Just as for number-line inequalities, my first step is tofind the "equals" part. For two-variable 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 thepoint where your book probably gets complicated, with talk of "test points" and such. But when you did those one-variable inequalities (like x < 3), you didn't bother with "test points"; you justshaded 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 theline); 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 theline, 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 yalone on one side of the inequality. Just as with plain old lines, you always want to "solve" the inequality for y on one side.
Graph the solution to 2x – 3y < 6.
First, I'll solve for y: