Graph the inequality y y y < 2 x 2^{x} 2 x .

Hey, everyone. So, hopefully, you had a chance to work this one out in your own. In this practice problem, we're gonna graph this inequality y is is less than 2 to the x power. Now remember, this is the x the equation of an exponential, equation. So this is a nonlinear inequality. But remember that we just use the same exact steps to graph nonlinear inequalities as we do any inequality. So let's go ahead and get started here. We're just gonna take a look at the first step here. Are we dealing with a solid or a dashed line? Really just comes down to the symbol over here. We have a less than symbol, so we're gonna be using a dash line. And we're gonna graph this just by switching the inequality symbol with an equal sign. So what does the graph of y equals 2 to the x look like? Now remember, just generally, exponentials will look kind of like this, But let's just go ahead and plot out some points here. Remember that the easiest points to plot out are gonna be, stuff like 0, because 2 to the 0 power will just get you a y value of 1. 2 to the 1 power will just be 2. So in other words, at x equals 1, you'll have a y value of 2. What about x equals 2? Well, that just means that you have 2 to the second power, and that will give you 4. So this is generally just gonna look like this. Now over here, it's gonna approach the sort of y, it's gonna approach the 0 points over here. That's the asymptote. So this formula is gonna look something like this. And remember, we're gonna have to use a dash line. So it's gonna look something like this. Alright? So this is gonna be the equation of y is equal to, or less than 2 to the x power. And now we just have to go go ahead and just test the point on either side by plugging, into the inequality, the x and y values. So I'm just gonna go ahead and pick, let's just say I'm just gonna use this point over here. So this is the point 1 comma 0. Remember, it could be x or y axis. And now we just have to plug these points into our inequality. Alright? So in other words, what this inequality is saying is, is 0 less than 2 to the one power? So is that a true statement? Well, 0 is, in fact, less than 2. 2 to the one power is just 2. So is 2 is 0 less than 2? It is. And so because I have a true statement over here, then we're going to shade the side that includes that point. So that was steps 23. Alright? So this point over here is definitely gonna be included within our region. So, actually, we just have to shade this entire region here that includes all of those points, all the, that that whole side of this inequality. Alright? So it's just gonna be looking like this. So very similar to a line, what you'll see is that lots of linear inequalities or sorry, nonlinear inequalities kind of obey the same rule as the linear inequalities. Generally, if you see a less than symbol, then really just gonna looking be looking to shade everything that's kind of below that graph, and that's exactly what we have here. Alright? So this is the inequality for y is less than 2 to the x power. Hopefully, that made sense, and thanks for watching.