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Video: Identifying Regions That Represent the Solutions to a System of Inequalities

Bethani Gasparine

Which regions of the graph contain solutions to both of the following inequalities? 𝑦 > −𝑥 − 4 and 𝑦 ≥ 2𝑥 − 4.

03:04

Video Transcript

Which regions of the graph contain solutions to both of the following inequalities: 𝑦 is greater than negative 𝑥 minus four and 𝑦 is greater than or equal to two 𝑥 minus four?

So these are inequalities, meaning they’re not equations; they don’t use equal signs. However, we need to use equations first to decide which lines go with our two inequalities. The equation of a line is 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 is the slope, the steepness of the graph, and 𝑏 is the 𝑦 intercept where it crosses the 𝑦-axis.

The slope is rise over run, meaning the change vertically divided by the change horizontally. If we go up, that’s positive. If we go down, that’s negative. If we go right, that’s positive. If we go left, that’s negative. So let’s first begin looking at our 𝑦-intercepts. They’re both at negative four, which can be found here, and there are actually only two lines that go through that point, so those are probably our two inequalities.

Let’s look at the slopes next. Our first slope has a negative in front of 𝑥; it’s like a negative one, which we can rewrite as negative one over one, so down one vertically and then right one horizontally, which would get us to this point. And if we would do it again, we’d get here, and so on. So this is the line that goes with 𝑦 is greater than negative 𝑥 minus four.

So why is that line dashed? Dashed lines are where greater than or less than is being used. Solid lines are when greater than or equal to or less than or equal to are used. So here we have greater than that’s being used; that’s why it’s dashed.

Next is shading. If we use greater than or greater than or equal to, we need to shade above that line. If we use less than or less than or equal to, we shade below that line. So since our first inequality is 𝑦 is greater than negative 𝑥 minus four, we need to shade above that line, almost thinking of being on top of the line, like you could walk on the line. So we will need to shade everything above that line.

So now let’s look at our other inequality. It has a slope of two or two over one. So starting at that negative four on the 𝑦-intercept, we need to go up two right one, which indeed works with this line. And it makes sense that this is a solid line because it’s a greater than or equal to.

So now when it comes to shading, it’s a greater than or equal to, so we need to shade above that line, which would be this way. Again, you would wanna be above it, like you could walk on top of it. You do not wanna be underneath it, where you could almost push up on it like it’s a ceiling.

So it says which regions of the graph contain a solution. So what are the solutions? Well, the solutions are where these two regions overlap, so we can see all of our solutions are in the regions of 𝐴 and 𝐷. Therefore, our final answer will be 𝐴 and 𝐷.