Graphing 2 Variable Inequalities - Problem 2
To graph a linear inequality, start by putting it into slope-intercept form; in other words, solve the inequality for y. Once it is in slope-intercept form, graph the inequality as you would a line in slope-intercept form y=mx+b. This means to first plot the y intercept (the "b" value) on the y-axis. Then use the slope (the "m" value) to plot a second point. If the inequality is "greater than" or "less than" (but not equal to), then the line connecting the points should be a dotted line. This means that the possible solutions to the inequality do not have values on the line. If the inequality is "greater than or equal to" or "less than or equal to", then the line connecting the points should be a solid line. This means that the possible solutions include the values on the line. Lastly, shade the appropriate region. For values that are "greater than" or "greater than or equal to" the line, shade the region above the line. For values that are "less than" or "less than or equal to", shade the region below the line.
I want to graph this inequality that has two variables but before I can approach it, I want to put this line into y equals mx plus b form so it's easier to graph. You don't have to, you could make a table, you could use y intercepts those are a couple other options that you guys are already know about. For me personally I prefer y equals mx plus b form.
So first thing I'm going to do is subtract 3x from both sides so that I have 2y is greater than or equal to -3x plus 8. Then I want y all by itself so I need to divide everything by two. Y is now greater than or equal to -3/2x plus 4, 8 divided by 2 is 4. Okay now that my line's in y equals mx plus b form I know how to graph it.
My first dot goes on the y axis at 4; from there I'm going to count the slope. So here we go my first dot goes on the graph on the y axis at 4, one, two, three, four. From there I want to count the slope which is down 3, right 2. So from there I'm going to go down 3, right 2 and then connect it. Okay, that's the first step, is to get my line on the graph.
Next thing I want to do is decide if this should be a solid line or dashy line and so the way I can tell it is looking at the inequality symbol. Since it says greater than or equal to I'm going to leave it as a solid line that's easy. Step two, check.
Next thing I need to do to finish my graph is to do some shading. I'm going to be shading half of the plane, either the half above the line or the half below the plane like that, you can make sound effects if you want to.
So I'm going to pick any point I want to that's not on the line. I usually use (0,0) then I'm going to substitute in those values for x and y see if I get a true statement. Let me go back to the original problem. Is it true that when I plug in (0,0) for x and y I get a true inequality 0 is bigger than 8? No that's okay, what that means is that don't shade here this tells me don't shade that half of the plane but instead shade the other half.
If this half gave me no that means that this half on top of the line will give me the yeses meaning every point above the line, above meaning out here, will be a solution to this inequality when I plug it in the x and y values.
This problem is a little tricky because you need to get it into y equals mx plus b form before you can graph it but once you do that it's pretty straight forward. So don't be intimidated if your homework problems look like this, where it's not already in y equals mx plus b form.