Welcome back, everyone. Let's continue our journey through systems of inequalities by taking a look at this example problem here. Here, we're told we're given these three equations, a, b, and c, and we're told to match each inequality with its graph. So we're not actually going to graph anything; we're just going to be matching up equations to graphs. Let's go ahead and get started here. So, I've got equation a, which is y>-3x+4. Then for equation b, we've got x+y<1. Equation c is -3x-y>=-4. Let's take a look at the first equation over here. y>-3x+4. Now, in order to visualize the graph, remember, the first thing you do is just pretend here that there is an imaginary equal sign. What does the graph of -3x+4 look like? It looks like an equation of a line with a y-intercept of 4. If you look through these two graphs, there are 2 equations with y-intercepts of 4. The graphs 2 & 3. And they almost look the same, except how the shading is in different areas, and one solid, and one is a dashed line. So which one does this equation correspond to? The next thing you should look at is the symbol that we see; because here we see a greater than symbol. Whenever you see a greater than symbol or a less than symbol without the solid bar underneath, that means we're dealing with a dashed line, not a solid line. So that means that without a doubt, equation a corresponds to graph number 3. So this corresponds to graph a. Alright?
And if you can see here, we've got the dashed line with a negative slope of 3, and we also can see that the shaded area is everything that's above that line, which is exactly what the shortcut for lines tells us. Notice how this symbol here is a greater sign, so we're going to shade everything that's above that line. Alright. So now let's take a look at equation b. x+y<1. So this may be hard to visualize what the graph looks like, so let's rewrite it in slope intercept form. I could just move the x to the other side by subtracting x, and this is going to be y<-x+1. So, same thing over here. We've got a y-intercept of positive one. And notice how what I said in the beginning or when we were dealing with the first equation is that these two lines have y-intercepts of 4. The only one that has a y-intercept of 1 is going to be this line. So this is definitely corresponds to equation b. Alright? And just to sort of double-check here, we can see that this equation tells us that we should be dealing with a dashed line because it's just a less than symbol, and that's exactly what we have. We have a dashed line. It's got a slope of negative one. And, also, what we can see is that because of the less than symbol, our shortcut tells us that we have to shade everything that's below that graph. So this is, without a doubt, the equation for the graph of b. And that just means that by default, this second graph over here is going to be the equation for c. But let's understand why that works.
So here, we've got this other sort of more complicated, nasty expression. One of the things that you can do here is you can sort of tell that all of these graphs or these numbers here have a negative sign. And so one thing you can do is you can actually flip all of those to positives, but then you have to flip the inequality symbol. So one way this sort of, like, simplifies is this actually just turns into 3x+y<=4. When you flip all the signs, you actually flip the symbol as well. And finally, what we can do is move the 3x to the other side, which you'll get is y<=-3x+4. That would be in slope-intercept form. So, again, notice how we still have that positive 4, of a y-intercept. We still have the same slope of negative 3. But now what happens is that we have a less than or equal to, so this should be a solid line, and we should shade everything that's underneath that line instead of above like we did for graph number 3. Alright? So that means that this is definitely the equation or the equation for this graph over here.
Alright. So that's it, folks. Let me know if you have any questions, and thanks for watching. I'll see you in the next video.