 In this video, we're going to solve inequalities involving power functions, particularly those with rational exponents. Now the way to solve an inequality, it doesn't matter what function is in play, you can first solve the corresponding equation. So we wanna take five times x minus two to the three halves power and set that equal to 135 and solve for it. To begin, I would divide both sides by five. We then get x minus two raised to the three halves power and that's then gonna equal well, five goes in 135 27 times. And so next we're gonna take the two thirds power of both sides, the two thirds power. On the left hand side, we're just gonna get x minus two. On the right hand side, we're gonna get the cube root of 27 squared, which the cube root of 27 is a three and three squared is equal to nine. We then add two to both sides and we see that our marker is gonna be x equals 11. And so this is the x-axis we now wanna consider. So we have our marker at 11. So either this entire interval, which would be 11 to infinity is part of the solution or it's not. And then the other one, negative infinity to 11, either that's part of everything or not. In which case we could plug these into the inequalities if you want to. We can also try to think of it in terms of graphing these things, what's going on here. So if we try to graph the function y equals five times x minus two to the three halves minus 135. So I just set the right hand side equal to zero, you subtract 135 from both sides. If we think about in terms of transformations, what's going on here. We have our standard function, y equals x to the three halves. So this is like a square root function that's been distorted. So graphically it would look something like the following, just roughly speaking, in terms of transformations, we've vertically stretched it by a factor of five, great. Then we shifted it to the right by two, that's great. And then we shifted it down by 135. In particular, because of these shifts, the right hand side of the graph should be positive and the left hand side of the graph is gonna be negative. And again, you could just plug in a test value, you could plug in something like x equals zero, what happens there? Plug in x equals 2,000, see what happens there. Test points are acceptable as well. So what I predict is the following, this should be above the x-axis and this should be below. And I should make an amendment to what I said before, I wouldn't go down to negative infinity. The domain here is as far as down as we could go would be, it looks like two. Because if we take something less than two, x equals two right here, you're gonna get a negative number and negative to a square root is gonna be imaginary. So it actually would go, the two possibilities were less than 11, which is two to 11 and greater than 11, which is 11 to infinity. And the graph suggests that it's gonna be positive here. And so our function would then be, we want all things that are greater than 11. Because our inequality is greater than or equal to, our solution would include the marker 11 towards infinity. And again, if you don't like this graphical approach, just plug in a specific number here. I would plug in something for which I expect to get a perfect square. Like for example, if I take the number six, take x equals six for example, you would end up with five times six minus two to the three halves. That gives us five times four to the three halves. The square root of four of course is two. So we get five times two cubed. That's gonna give us five times two cubed is eight. And so five times eight is 40. 40 is not bigger than 135. So we see that having number less than 11 didn't work. So you can use test points as well. If you don't like this graphical approach, whichever one you want, either one's gonna work. Personally I like graphing, but that's because I'm more familiar with the graphing. If you are very much a beginner, the graph might seem less than obvious. And so a test point might be a simpler way of approaching it here. So to solve this inequality, we're gonna take three times x plus one, raise it to the two thirds power minus one and set that equal to 107. Now be aware that in the future, we're gonna consider graphing the function three times x plus one to the two thirds minus 108. So you can always set this equation equal to zero. And that's the function you wanna graph to see if you're above or below the x-axis. We're gonna be looking for things that are going to be below the x-axis in just a little bit. But let's solve this equation here to find the marker. I'm gonna add one to both sides. So we get three times x plus one to the two thirds that's equal to 108. We're gonna divide both sides by three. So we get x plus one to the two thirds is equal to 108 divided by three, which is equal to 36. And then taking the, so we divided both sides by three here. We added one both sides here. And this side we're gonna take the three halves power. So this gives us x plus one, x plus one is equal to the square root of 36 to the cubed. Oh, wait, we just square rooted both sides. We probably need to have two solutions, plus or minus one, there you go. The square root of 36 is going to be six. So then we have to cube six, which gives us 216 plus or minus. So we wanna think of in terms of our x-axis, we have these two markers, one at 216 and one at negative 216. So come back to the original graph, the original function right here. What is it that we're trying to do to this thing? The original function because we're squaring it, even though there's a one third, it's gonna kind of look something like this. Is this a perfect picture? No, but this will be sufficient. This graph, we're gonna vertically stretch it, okay? We shift it to the left by one, and then we're gonna shift it down by a negative, negative, we're gonna shift it down by 108. So something like this is gonna happen. This is kind of like our beak versus the wings type portion right here. There's the beak, which is gonna be below the x-axis, and then there's gonna be the wings that are above the x-axis. Because we're looking for things below the x-axis, I need to be selecting the beak on this bird right here. And actually it occurs to me that I didn't actually finish solving this one thing right here, 216's aren't actually correct. Cause what we saw here is that x plus one is equal to plus or minus 216. I need to still subtract one from both sides. So we get x equals negative one plus or minus 216, which tells us that we are going to get actually 215 or negative 217. In terms of the geometric analysis, that makes no bit of difference whatsoever, but it does have an effect on us. If we want the beak, we are going to grab the things between negative 217 and 215. And looking back at the original inequality, equality was not allowed there. So we're just gonna get the points between the markers there. That's just the beak of our bird. And so our solution would look like negative 217, 215. Which, if you don't feel comfortable with the graphing, like on the other example, you could plug in test values. Zero's a good number right here. Maybe something like 10 million. You know, it doesn't, it's just to be a big number. We have to get the, you can pick something smaller if you want like three, negative 300. Just pick some points to plug in the function that will simplify very nicely. By all means, find things that are gonna be perfect squares for this situation, our perfect cubes. And this helps us solve, this demonstrates a technique how we can solve inequalities involving rational exponents. And this would work for power functions in general as well.