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