In Exercises 1–26, graph each inequality. y2^x

Question

In Exercises 1–26, graph each inequality. y>2^x

Accepted Answer

Welcome back. I am so glad you're here. We are given the inequality. Why is less than four raised to the X and were asked to graph it. I'm going to start off giving us a little bit more room to do our graphing. And then I am going to draw a rough sketch of a coordinate plane will have a vertical Y axis, a horizontal X axis. They come together at the origin in the middle. And now in order to graph our inequality, why is less than four raised to the X? We first need to turn this into an equality we're going to set why equal to four raised to the X. And then the next thing we do is we check and see if R equals is going to be included. And this is just why is less than four raised to the X? It's not why is less than or equal to. So when we draw our line for Y equals four raised to the X, we are going to need to draw it as a dotted line. Alright, now we recall from previous lessons that when we have a function that is why equals a number raised to the X that is going to have a horizontal asam tote along the X axis starting from negative infinity. So it's going to be as close to it can get without touching and then it's going to have a Y intercept at 01. So it'll come up like this and then it'll pretty decently start heading up to positive infinity for both X and Y values. And there is our line Y equals or function line Y equals four raised to the X. Now going back to our inequality. Where is it true that why is less than for raised to the X? Well, in order to discern that we choose test points, we're going to choose one test point that is above this function line. And then we're going to choose one test point that is down below this function line. And so to choose our test points, we can choose um one test point at, let's say this is 05. Oops got fancy zero five for one test point and then one test point that is below, let's choose zero, negative five. And so now we plug both of those values into the inequality and see which one's true. So for 05, ry value is five, we got five is less than four raised to the R X value is zero. Well, simplifying on the right hand side, anything raised to the zero power is one. So four raised to the zero is one, is it true that five is less than one? It's not true. So that inequality does not work. Now, let's test the other one just to make sure we've done that correctly. Now, plugging in our Y value is negative five is less than four raised that we still have zero for our X value. We know that the right hand side simplifies to one. So is it true that negative five is less than one? Yes, it is. That's true. So we get to shade in this whole area, this direction of the function line and this area up above into the left is not included. All right. So now we've got this inequality shaded in, let's head back up to the top and see which answer choice matches with my lovely green blob that I have drawn. Sure it will look precisely like my lovely green blob. Okay. All right. We want to have one that is heading toward the X axis, heading toward negative infinity and heading up toward positive infinity on the right hand side. And that looks like our, our dotted lines for C and D match. And now which one has the green blob down into the right of the function line. That is answer choice C, that one has it shaded in down into the right of the function line. So answer choice C works well done. We'll catch you on the next one.

In Exercises 1–26, graph each inequality. y>2^x

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