Question Video: Graphing Combinations of Inequalities Mathematics

The shaded area of the following figure represents the solution set of the inequalities _.

03:14

Video Transcript

The shaded area of the following figure represents the solution set of the inequalities blank.

Well, the first thing we want to do with any problem like this is identify the equations of each of our straight lines. Now, in this problem, it should hopefully be quite straightforward because all we have are horizontal and vertical lines. So, what we know about a horizontal and vertical line in their equation is that a horizontal line is going to be 𝑦 is equal to a constant. That’s because the 𝑦-coordinates along that line would not change. And a vertical line is going to be 𝑥 is equal to a constant. And once again, that’s because the 𝑥-coordinates along that line will not change.

Well, if we start with our vertical lines, our first vertical line is going to be 𝑥 equals two. And that’s because we can see that the line goes through the two on the 𝑥-axis. So, all of the 𝑥-coordinates on this line will be two. Then, our next line is going to be 𝑥 equals five. So, that’s our next vertical line. And then, our first horizontal line will be 𝑦 equals six cause you see that it crosses the 𝑦-axis at six. And then we move on to our final line, 𝑦 equals three.

Okay, great. So, we have the equation of all of our lines in our graph. Well, now that’s all well and good. But what we’re looking for is, in fact, inequalities. So, let’s remind ourselves a little bit about how we graph inequalities. Well, if we remind ourselves how we draw inequalities on a graph, then if we’ve got a dashed or a dotted line, this is gonna be greater than or less than. However, if we have a solid line, it’s gonna be greater than or equal to or less than or equal to.

So, if we take a look at our graph, we can see that all the lines are in fact solid lines. So all our inequalities will include the “or equal to.” So, first of all, let’s have a look at the shaded region, and let’s see how it’s bound in the 𝑥-axis, so between our vertical lines. So, we can see it’s to the right of 𝑥 equals two, but to the left of 𝑥 equals five. So, we can see that 𝑥 is gonna be greater than or equal to two because it’s to the right of the line 𝑥 equals two — and remember, it’s “or equals to” because we’ve got that solid line — but less than or equal to five because it is to the left of the line 𝑥 equals five.

Now, we could’ve written this as two single inequalities. However, it works better and it’s a neater way of writing it as a double inequality because then we can see that 𝑥 must lie between two and five, but it could also include two or five. Okay, that’s 𝑥 dealt with.

Now, let’s have a look at the 𝑦-part of our inequality. Well, if we look at our horizontal lines, we can see that the shaded area is above the line 𝑦 equals three but below the line 𝑦 equals six. So, therefore, we would say 𝑦 is greater than or equal to three and less than or equal to six.

So, therefore, we can say that the shaded area of the figure represents the solution set of the inequalities 𝑥 is greater than or equal to two but less than or equal to five. And 𝑦 is greater than or equal to three but less than or equal to six.

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