Video: Determining the System of Inequalities Represented by a Given Graph

Chris O’Reilly

State the system of inequalities whose solution is represented by the following graph.

03:53

Video Transcript

State the system of inequalities whose solution is represented by the following graph.

Okay, the first thing we look at, we’re looking for an inequality whose solution is represented by the following graph because of the fact that we can actually say that 𝑥 and 𝑦 are both gonna be greater than or equal to zero. We can actually say that because if we look, we’ve got this quadrant of the axis, but also we have the arrows on our lines and on our axes, which say that it’s continuing on. So therefore, we can say that 𝑥 is greater than or equal to zero and 𝑦 is greater than or equal to zero.

So great! We have our first two inequalities whose solutions is represented by the following graph. Now, we’re gonna take a look at this section and see which inequality is gonna be represented by this area. Before we do, we’re gonna have a look at this little reminder that tells us that if we have a solid line, it’s gonna be equal to, greater than or equal to, or less than or equal to. However, if it’s a dashed line or nonsolid line, then it’s equal to greater than or less than. And what this actually means is that this one wouldn’t include the values themselves that the line is on. However, if it’s a solid line, it would include those values.

If we take a look at our region, first of all, we can see that 𝑦 is greater than or equal to two. And that’s because if we look at the bottom it’s two and it’s going upwards with the region and it’s a solid line, so therefore it’s greater than or equal to. But it’s also less than or equal to six, as again the line is on six and it’s a solid line. Great! So fantastic, we’ve now got another inequality whose represented by the graph.

Let’s move on to another region. Well if we take a look at this region, we can see that 𝑥 is gonna be greater than or equal to three again cause we have our solid line at three and we can actually see that the region is to the right-hand side of that. So it’s gonna be greater than or equal to three. But it’s gonna be less than six. And this time, pay attention to the fact that it’s less than. And it’s less than not less than or equal to because we have a dashed line this time on the line of six, where 𝑥 is equal to six. So therefore, we can say that 𝑥 is greater than or equal to three and less than six.

Right, so that’s another one of our inequalities whose represented by the graph. Let’s move on to the next one. Well, our next region is this triangular region here. And in order to find out what the inequality that’s represented by this region is, we want to work out the equation of the line. Well, if we look at the line, we can actually see that 𝑥 plus 𝑦 is gonna be equal to eight. For example, we’ve got point where when 𝑥 is equal to zero, 𝑦 is equal to eight. So zero plus eight equals eight. And if we look at the point where 𝑥 is equal to three, we can see that 𝑦 is equal to five. So again, that would satisfy our equation because three plus five is equal to eight.

Okay, great! So we now know the equation of the line. So therefore, we can say that our inequality would be 𝑥 plus 𝑦 is less than or equal to eight. And it’s less than or equal to eight because it’s wanting all the values actually to the left-hand side of our line that we found the equation of. So as we can see, it’s less than eight. And also, we have less than or equal to because it’s again a solid line.

And we can just double check that. So I picked a point here. So I picked the point when 𝑥 equals three, 𝑦 equals two. Well, if we had three plus two, we’re not gonna get eight. And we are in fact gonna get an answer that’s less than eight. So great! Yes, 𝑥 plus 𝑦 is less than or equal to eight would be a correct inequality whose satisfied by this region.

So therefore, we can say that the system of inequalities whose solution is represented by the following graph are 𝑥 is greater than or equal to zero, 𝑦 is greater than or equal to zero, 𝑦 is greater than or equal to two or less than or equal to six, 𝑥 is greater than or equal to three or less than six, and finally 𝑥 plus 𝑦 is less than or equal to eight.

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