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

Chris O’Reilly

Which regions on the graph contain solutions to both of the following inequalities? [A] 𝑦 >𝑥 [B] 𝑦 ≤ 2𝑥 − 4

03:21

Video Transcript

Which regions on the graph contains solutions to both of the following inequalities, 𝑦 is greater than 𝑥 and 𝑦 is less than or equal to two 𝑥 minus four?

To solve this problem, we’re actually going to start with this inequality that says that 𝑦 is greater than 𝑥. So as you can see on our graph, we’ve already got the line 𝑦 equals 𝑥. Well we’re interested in all the regions where 𝑦 is greater than 𝑥. So we actually want everything that’s this side of the line. So therefore, we can eliminate all these regions as these regions are all where 𝑦 is less than 𝑥.

Okay, great! Let’s move on to the next inequality. well our next inequality states that 𝑦 is less than or equal to two 𝑥 minus four. And what I wanna do actually just to see which one of the lines on our graph represents two 𝑥 minus four.

I’m actually gonna substitute some values in. The values that we’re gonna substitute in for 𝑥 are 𝑥 is equal to zero, two, and four. And I am gonna put those values in just so we’ve got enough to show exactly which line is going to be equal to our 𝑦 is equal to two 𝑥 minus four.

Well first of all, if I substitute in 𝑥 is equal to zero, I’m gonna get 𝑦 is equal to zero minus four. So therefore, 𝑦 is gonna be equal to negative four. So then our first point will be zero, negative four. So I’ve now marked that on the graph so we can see where that is.

Okay, substituting our next value of 𝑥, so 𝑥 is equal to two, we get 𝑦 is equal to two multiplied by two minus four, which actually give a 𝑦-value of zero. So now we can mark that on our graph. Okay, great! That’s now on there so it looks like we’ve identified the line that’s going to be 𝑦 is equal to two 𝑥 minus four. But we’ll just do our final value just to make sure.

So in substituting, 𝑥 is equal to four. So we get 𝑦 is equal to two multiplied by four minus four, which is equal to four. So great! Our final point is gonna be at the coordinates four, four. Okay, with that marked on the graph, it’s clear that we’ve now identified our line, 𝑦 equals two 𝑥 minus four.

So now we’ve got 𝑦 is equal to two 𝑥 minus four. We can actually go back to our inequality. And what we want, we want it when 𝑦 is less than or equal to two 𝑥 minus four. So therefore, we can eliminate all the regions that are actually going to be greater than two 𝑥 minus four.

So we’re left with one region. And that region is 𝐺. So we can say that region 𝐺 contains solutions to both of our inequalities. And yes, we’ve actually reached our final answer. But I just want to double check. So I want to bring your attention to our inequality signs.

For our first inequality, it says that 𝑦 is greater than 𝑥. So therefore, it’s just greater than. So it’s not greater than or equal to. So what we want is a dashed line. And if we actually take a look at the line 𝑦 equals 𝑥, yes we can see it’s a dashed line. So that’s great!

And for our second inequality, it says that 𝑦 is less than or equal to two 𝑥 minus four. And because it’s less than or equal to, we actually want a solid line. So look at the line. Yes! Great! It’s a solid line. So yep, we can be happy that region 𝐺 contains solutions to both our inequalities.

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