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Two-variable inequalities from their graphs

Sal is given a graph and he analyzes it to find the two-variable inequality it represents. Created by Sal Khan and Monterey Institute for Technology and Education.

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Video transcript

Write an inequality that fits the graph shown below. So here they've graphed a line in red, and the inequality includes this line because it's in bold red. It's not a dashed line. It's going to be all of the area above it. So it's all the area y is going to be greater than or equal to this line. So first we just have to figure out the equation of this line. We can figure out its y-intercept just by looking at it. Its y-intercept is right there. Let me do that in a darker color. Its y-intercept is right there at y is equal to negative 2. That's the point 0, negative 2. So if you think about this line, if you think about its equation as being of the form y is equal to mx plus b in slope-intercept form, we figured out b is equal to negative 2. So that is negative 2 right there. And let's think about its slope. If we move 2 in the x-direction, if delta x is equal to 2, if our change in x is positive 2, what is our change in y? Our change in y is equal to negative 1. Slope, or this m, is equal to change in y over change in x, which is equal to, in this case, negative 1 over 2, or negative 1/2. And just to reinforce, you could have done this anywhere. You could have said, hey, what happens if I go back 4 in x? So if I went back 4, if delta x was negative 4, if delta x is equal to negative 4, then delta y is equal to positive 2. And once again, delta y over delta x would be positive 2 over negative 4, which is also negative 1/2. I just want to reinforce that it's not dependent on how far I move along in x or whether I go forward or backward. You're always going to get or you should always get, the same slope. It's negative 1/2. So the equation of that line is y is equal to the slope, negative 1/2x, plus the y-intercept, minus 2. That's the equation of this line right there. Now, this inequality includes that line and everything above it for any x value. Let's say x is equal to 1. This line will tell us-- well, let's take this point so we get to an integer. Let's say that x is equal to 2. Let me get rid of that 1. When x is equal to 2, this value is going to give us negative 1/2 times 2, which is negative 1, minus 2, is going to give us negative 3. But this inequality isn't just y is equal to negative 3. y would be negative 3 or all of the values greater than negative 3. I know that, because they shaded in this whole area up here. So the equation, or, as I should say, the inequality that fits the graph here below is-- and I'll do it in a bold color-- is y is greater than or equal to negative 1/2x minus 2. That is the inequality that is depicted in this graph, where this is just the line, but we want all of the area above and equal to the line. So that's what we have for the inequality.