WEBVTT
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Which inequality has been graphed in the given figure?
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So in this problem, what we are looking at is the shaded region below the dashed line.
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So the first thing we want to do in any problem like this is find the equation of the straight line that is represented by, in this case, a dashed line.
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Well, in order to do this, what we’re going to do is use the general form for the equation of a straight line.
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And that is 𝑦 equals 𝑚𝑥 plus 𝑐, where 𝑚 is the slope and 𝑐 is the 𝑦-intercept.
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Well, straightaway, we can see that 𝑐, our 𝑦-intercept, is gonna be negative two because that’s where it crosses the 𝑦-axis.
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Well, next, we want to find our slope.
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And the way that we could find our slope is by thinking about it as the change in 𝑦 over the change in 𝑥.
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Or we could also think about it as the rise over the run.
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And if we want to think about that in a more formal way, there is a formula which is 𝑚 is equal to 𝑦 sub two minus 𝑦 sub one over 𝑥 sub two minus 𝑥 sub one, where our line passes through the points 𝑥 sub one, 𝑦 sub one and 𝑥 sub two, 𝑦 sub two.
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So this just means the change in 𝑦 over the change in 𝑥.
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So in order to find the slope, what we want to do is find two easily readable points on our line.
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It wouldn’t matter where they are because it’s a straight line, so the slope will not change.
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Well, we could see with the two points that we’ve chosen on our graph that the change in 𝑥 is gonna be three because there are three units along and the change in 𝑦 is gonna be negative one.
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That’s because it’s going down one.
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So therefore, we could say that the slope is gonna be equal to negative a third or negative one over three, and this is what we’d expect.
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And that’s because if we have a line that goes down to the right, then we expect the slope to be negative.
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If it goes up to the right, we expect the slope to be positive.
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So therefore, we can say that the equation of the line represented by the dashed line is 𝑦 equals negative a third 𝑥 minus two.
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Now, have we solved the problem here?
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Well, not quite, because what we’re looking for is the inequality that’s been graphed in the given figure.
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Well, what we’re gonna do is remind ourselves of the significance of the dashed line.
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Well, if we’ve got a dashed line, it means that the inequality is gonna be greater than or less than.
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However, if we’ve got a solid line, it’s gonna be greater than or equal to or less than or equal to.
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Well, as the region we’re interested in, because it’s the region that’s shaded in, is the region below the line, we know that’s gonna be 𝑦 is less than.
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And because we’ve got a dashed line, it’s just gonna be less than; it’s not gonna be less than or equal to.
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So therefore, we can say the inequality that’s being graphed is 𝑦 is less than negative a third 𝑥 minus two.