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Write an inequality that fits the graph shown below.

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So here they've graphed a line in red, and the inequality

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includes this line because it's in bold red.

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It's not a dashed line.

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It's going to be all of the area above it.

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So it's all the area y is going to be greater than or

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equal to this line.

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So first we just have to figure out the

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equation of this line.

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We can figure out its y-intercept just

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by looking at it.

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Its y-intercept is right there.

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Let me do that in a darker color.

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Its y-intercept is right there at y is equal to negative 2.

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That's the point 0, negative 2.

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So if you think about this line, if you think about its

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equation as being of the form y is equal to mx plus b in

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slope-intercept form, we figured out b is equal to

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negative 2.

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So that is negative 2 right there.

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And let's think about its slope.

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If we move 2 in the x-direction, if delta x is

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equal to 2, if our change in x is positive 2, what is our

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change in y?

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Our change in y is equal to negative 1.

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Slope, or this m, is equal to change in y over change in x,

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which is equal to, in this case, negative 1 over 2, or

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negative 1/2.

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And just to reinforce, you could have done this anywhere.

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You could have said, hey, what happens if I go back 4 in x?

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So if I went back 4, if delta x was negative 4, if delta x

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is equal to negative 4, then delta y is

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equal to positive 2.

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And once again, delta y over delta x would be positive 2

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over negative 4, which is also negative 1/2.

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I just want to reinforce that it's not dependent on how far

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I move along in x or whether I go forward or backward.

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You're always going to get or you should always get, the

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same slope.

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It's negative 1/2.

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So the equation of that line is y is equal to the slope,

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negative 1/2x, plus the y-intercept, minus 2.

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That's the equation of this line right there.

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Now, this inequality includes that line and everything above

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it for any x value.

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Let's say x is equal to 1.

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This line will tell us-- well, let's take this point so we

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get to an integer.

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Let's say that x is equal to 2.

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Let me get rid of that 1.

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When x is equal to 2, this value is going to give us

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negative 1/2 times 2, which is negative 1, minus 2, is going

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to give us negative 3.

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But this inequality isn't just y is equal to negative 3.

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y would be negative 3 or all of the values greater than

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negative 3.

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I know that, because they shaded in this

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whole area up here.

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So the equation, or, as I should say, the inequality

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that fits the graph here below is-- and I'll do it in a bold

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color-- is y is greater than or equal to

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negative 1/2x minus 2.

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That is the inequality that is depicted in this graph, where

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this is just the line, but we want all of the area above and

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equal to the line.

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So that's what we have for the inequality.

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