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

Tim Burnham

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

05:11

Video Transcript

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

And we’ve been given a graph with two lines; let’s call them one, which is the dotted line, and two, which is the solid line. Now for region one, we’re talking about points in the red region here, so above that line. And for region two, we’re talking about this yellow region here, so points above this line. And then this region I’ve shaded here is the intersection of those two regions.

So with two lines and two corresponding regions, we’re looking for two inequalities. And for the first one, we’ve got a dotted line. So points on the line are not included in the region if you got a dotted line.

And that implies that we’re talking about a strict inequality, so 𝑦 is greater than something or 𝑦 is less than something. And because we’re looking at points which are above that line, in fact, in our case we’re gonna have a 𝑦-is-strictly-greater-than-type inequality.

Well first of all, let’s work out the equation of that line. And to do that, we need to know the slope and the 𝑦-intercept. So if I take a point on that line and then I increase the 𝑥-coordinate by one, the corresponding 𝑦-coordinate on the line would decrease by three.

And that happens all the way down the line; increase the 𝑥-coordinate by one, the 𝑦-coordinate goes down by three; increase the 𝑥-coordinate by one, the 𝑦-coordinate goes down by three So the slope is negative three. Now we need to think about the 𝑦-intercept. And that means where does it cut the 𝑦-axis. When my 𝑥-coordinate is zero, what’s the corresponding 𝑦-coordinate? Well in this case, it’s negative eight.

So my 𝑦-intercept is negative eight. Now the general equation of a straight line is 𝑦 is equal to the slope times 𝑥 plus the value of the 𝑦-intercept.

And with a slope of negative three and a 𝑦-intercept of negative eight, I can put those values into my general equation and I’ve got 𝑦 is equal to negative three 𝑥 plus negative eight and would normally write that as 𝑦 equals negative three 𝑥 take away eight.

Now that equation tells us about all of the 𝑦-coordinates on that line, and we’re interested in that red region and all the points which are in fact above the line not on it. So the 𝑦-coordinates of points in this region up here are all greater than the values on the line.

So for instance when 𝑥 equals zero, point on the line has a 𝑦-coordinate of negative eight. But we’re in our region; we’re interested in all these 𝑦-coordinates all the way up here, which are greater than negative eight.

So we can represent that red region with this inequality. All the 𝑦-coordinates are greater in the red region than their 𝑥-coordinate times negative three and then take away eight from that.

Okay, let’s go and work out the inequality for the yellow region. Well equation two has got a solid line, and that tells us that points on the line are included in the region.

So we’re looking for an inequality which is something like 𝑦 is greater than or equal to something or 𝑦 is less than or equal to something. And again in this particular case, all the points that we’re interested in are above that line; the yellow region is above the solid line , so it’s gonna be 𝑦 is greater than or equal to something.

So let’s go on and work out the equation of that solid line. So again we need to work out the slope and the 𝑦-intercept And for the solid line, every time I increase my 𝑥-coordinate by one, my 𝑦- coordinate goes up by three. So my slope is positive three.

And the 𝑦-intercept, when the 𝑥-coordinate is zero, the corresponding 𝑦- coordinate is three; that’s where it cuts the 𝑦-axis. So the 𝑦-intercept is also positive three

And again, we can now use that information to write out the equation of that second line. With a slope of positive three, we’ve got 𝑦 equals three 𝑥 and the 𝑦-intercept is also positive three. So the equation representing points on that solid line is 𝑦 equals three 𝑥 plus three.

So for that particular line, for that equation, if 𝑥 was equal to one, 𝑦 on the line would be equal to six. But our region not only includes that point, it includes all the points above it in the region. So 𝑦 can be equal to three 𝑥 plus three, but it can also be greater than that.

𝑦 is greater than or equal to three 𝑥 plus three. So points in the red region follow this inequality; points in the yellow region follow this inequality; and points in this region here follow both of those inequalities.

So our answer consists of both of those inequalities: 𝑦 is greater than three 𝑥 plus three and 𝑦 is greater than negative three 𝑥 take away eight.