Graphing two-variable inequalities
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We're asked to graph the inequality y is less than 3x plus 5. So if you give us any x-- and let me label the axes here. So this is the x-axis. This is the y-axis. So this is saying you give me an x. So let's say we take x is equal to 1 right there. 3 times 1 plus 5. So 3 times x plus 5. So 3 times 1 is 3 plus 5 is 8. So one, two, three, four, five, six, seven, eight. This is saying that y will be less than 8. y will be less than 3 times 1 plus 5. So the y-values that satisfy this constraint for that x are going to be all of these values down here. Let me do it in a lighter color. It'll be all of these values. For x is equal to 1, it'll be all the values down here, and it would not include y is equal to 8. Y has to be less than 8. Now, if we kept doing that, we would essentially just to graph the line of y is equal to 3x plus 5, but we wouldn't include it. We would just include everything below it, just like we did right here. So we know how to graph just y is equal to 3x plus 5. Let me write it over here. So if I were to write y is equal to 3x plus 5, we would say, OK, 3 is the slope. Slope is equal to 3, and then 5 is the y-intercept. Now, I could just graph the line, but because that won't be included in the y's that satisfy this constraint, I'm going to graph it as a dotted line. So we'll start with the y-intercept of 5. So one, two, three, four, five. That's the y-intercept. And the slope is 3. So if you go over to the 1, you go up 3. So let me do that in that darker purple color. So it'll look like this. It will look like that. That point would be on it. That point would be on it. If you go back, you're going to go down by 3. So that point will be on it, that point, and that point, and I'll just connect the dots with a dotted line. That dotted line is the graph of y is equal to 3x plus 5, but we're not going to include it. So that's why I made it a dotted line because we want all of the y's that are less than that. So for any x-- so you pick an x. Let's say x is equal to negative 1. If you evaluate 3x plus 5 for that x, you'd get here. But we only care about the y's that are strictly less than that. So you don't include the line. It's everything below it. So for any x you pick, it's going to be below that line. You take the x, go up to that line and everything below it. So for all of the x's, it's going to be this entire area. Let me draw it a little bit neater than that. It's going to be this entire area that's under the line. I'll do it in this orange. It's a little bit easier to see. So this entire area under the line is y is less than 3x.