Which inequality has been graphed in the given figure?
So in this problem, what we’re looking to do is identify which inequality would describe this area to the left of our dashed line. So it’s the area colored in blue. Well, to enable us to do that, the first thing that we want to do is to find the equation of the dashed line itself. Well, to help us do that, we’re gonna use this formula for the equation of a straight line. And that is that 𝑦 is equal to 𝑚𝑥 plus 𝑐, where 𝑚 is the slope or gradient and 𝑐 is our 𝑦-intercept.
Well, straightaway, we can actually see what 𝑐 is because our 𝑦-intercept is negative four ’cause we can see that the dashed line crosses the 𝑦-axis at negative four. So we have 𝑐 is equal to negative four.
So now, the next thing we need to do is find the slope of our line. And the reason we want to do that is because then we’ll have 𝑚 and 𝑐, and we can substitute them into our formula to give us the equation of our straight line. Well, in order to find the slope, what we need to do is pick two points on our line. Because it’s a straight line, we can pick any two points on the line. But the best thing to do is pick two points where it’s easy to read off what the coordinates are. So, for instance, we already had one here because that was at zero, negative four. And then the next point we’ve got here is at two, zero.
Well, the slope of a line is the change in 𝑦 over the change in 𝑥, sometimes known as the rise over the run. Or if we want to think about this more formally, we have 𝑚, our slope, is equal to 𝑦 sub two minus 𝑦 sub one over 𝑥 sub two minus 𝑥 sub one, so the change in 𝑦 over the change in 𝑥. And this is when we’ve got two points on our line: 𝑥 sub one, 𝑦 sub one and 𝑥 sub two, 𝑦 sub two. However, in our example, we’re not going to substitute into the formula because if we take a look at our graph, then we can see quite clearly that we’ve got a change in 𝑥 of two, because we go from zero to two. And we’ve got a change in 𝑦 of four, and that’s because we go from negative four to zero.
So therefore, we can say that our slope is gonna be equal to four over two, which is gonna give a slope of two. Okay, great, so now we have the slope and the 𝑦-intercept. So now what we’re gonna do is substitute our values in to give the equation of our straight dashed line. And when we do that, what we’re gonna get is 𝑦 is equal to two 𝑥 minus four.
Well, at this point, we think, well, have we finished? Well, the answer is no, because what we’re trying to do is find the inequality that’s been graphed in the given figure. And as we said earlier, it’s this blue region that we’re looking to find. So the first thing we need to do if we’re going to represent our shaded area with an inequality is remind ourselves what we use when we’re dealing with inequalities. So the notation we’ve got, if we have a dashed line, then this means it’s gonna be greater than or less than. However, if we had a solid line, it’d be greater than or equal to or less than or equal to.
So therefore, as the blue shaded region is in fact where 𝑦 is greater than 𝑚𝑥 plus 𝑐, we can say that the inequality that has been graphed in the given figure is 𝑦 is greater than two 𝑥 minus four. And the reason it’s just greater than not greater than or equal to is because, as we’ve just explored, we have a dashed line, not a solid line.