 I'd like to look at a few examples of solving rational inequalities. So our first example is given here, it's just a basic rational function. And the two main properties to remember and to think about when you're solving a rational inequality are the x intercepts of the function and the vertical asymptotes of the function. So if you think back to the properties that we talked about in our past class, the x intercepts always come from when the numerator is equal to zero. So in this case, our x intercept will just be at x equals four. And the vertical asymptotes come from when the denominator is equal to zero. And so in this case, our vertical asymptote will be just at x equals negative one. Now the reason that we want to look at the x intercepts in vertical asymptotes is because when we're solving this, we're comparing it to zero. And those are the only two spots where the function will either equal zero or jump over the x axis or go from positive to negative because of the asymptote. So we know an x intercept at four means our graph is actually going to go through that value at four. But at negative one, it's a vertical asymptote and so on our graph, we know it's never actually going to hit negative one. So that just gives a little rough idea of what our graph is going to look like. Now I think the easiest way to come up with what the rest of your graph will look like is going to be to type the equation into your calculator. So why don't you take a moment, pause the video, and type it into your calculator to see if you can finish up the sketch. Now when I sketch to this graph on the left here, I had a piece going like that and then on the right of that asymptote, it goes right through that zero and then goes down along negative one. So that is a rough sketch of the graph. Now from this graph, we want to compare when it is greater than or equal to zero. So greater than or equal to zero, again, is above the x axis. So we're thinking of it as if our number line is the x axis. And so if I start on the left, I'll trace along the graph because this is greater than or equal to zero, then I have to jump down because it's a vertical asymptote and this from negative one to four is below zero but then at four it's equal to zero so I'll include that and then continue on to the right on the graph. So everything in yellow has to be included in our solution. So again if we read this from left to right, from the left here it starts at negative infinity. So my interval will start from negative infinity all the way until negative one but it doesn't include negative one because it's an asymptote. So we use a parenthesis there and then in my highlighting it picks up again here at four and then continues to infinity. So the four has a bracket because it can equal zero and then to infinity at the end. So the solution here just contains two different intervals. Let's look at one more example that is slightly more complicated. So just like before we want to start out by finding two key properties. The x-intercepts which come from when the numerator is equal to zero. So here they occur at negative one and three. The vertical asymptotes which come from when the denominator is equal to zero. In this case we will have vertical asymptotes at x equals four and x equals negative five. So if we make a rough sketch again of our graph we can label these values on the graph. So I'll put negative one right here, that's an x-intercept so it will go through that point. Three here, another x-intercept, four right next to it is a vertical asymptote and then negative five down here is also a vertical asymptote. So we need to have a piece of graph for each part of the function. So try pausing the video and typing it into your calculator again to get what the graph will look like. When I initially graphed mine on the standard window I get a graph that looks like this in the middle. Notice it's just hitting at negative one and turning around and then I also see that there's a piece up here that goes along our vertical asymptote at negative four. But what you can't forget is there is a section of the graph here going down. So even if you can't see it on the window there still has to be some graph there. So zoom out if you need to in order to see what it looks like but when I did that I saw that there's a piece of the graph looking like that. So it will go right along that vertical asymptote at negative five. So now that I have my graph I can compare for my inequality. So notice I want it to be less than or equal to zero which is below the x-axis. So if I start from the left I'll trace along, this is below, so I'll include that then it jumps above, it can equal zero so I'm going to have to include this point at negative one and then I continue tracing. I include this point at three and it goes below until we get to four and then the rest is above. So there are three main sections to look at here where it is less than or equal to zero. So I'm running out of room a little bit to write the solution here but I'll try to fit it in the middle. If I describe this starting on the left this is negative infinity and then we go all the way to negative five where it's below. So negative infinity to negative five all with parentheses because negative five is an asymptote. Then it's equal again right here at negative one. We just include that one value and then our third piece starts at three and it includes three because it can equal zero all the way to four. So just a small piece there and four is going to have a parentheses because it is an asymptote. So there are three sections of the graph here where it is less than or equal to zero.