In Exercises 39–45, graph each inequality. y ≤ 2^x

Question

Accepted Answer

Everyone in this problem, we are asked to draw the graph of the given inequality in the rectangular coordinate system. The graph we're given is why greater than or equal to E to the X. And when it says rectangular coordinate system. Okay, if you look at the graphs of the possible answer choices here, what you'll see is that the horizontal units? Okay. One unit is wider than the one vertical unit. Okay, so the little boxes look like rectangles. Um because along the X axis there more spaced out than along the y axis. Okay, so that's all that means by rectangular coordinate system. So let's go down here. Give ourselves some space to draw this out. I'm just going to rewrite the inequality we were given. So we were given why is greater than or equal to E to the X. Now, when we're asked to graph an inequality, the first thing we want to do is graph the equality. Okay, here we know that the equality is included because we have greater than or equal to the equality is included. So let's go ahead and graph that. So we're gonna graph y is equal to E to the X. Okay. And what can be helpful for graphing is just to find the X and Y intercept first. Just can help guide the graph. So if we're looking for the X intercept. Okay, this is gonna be when y is zero. So we get zero is equal to E to the X. Okay, well a number to the exponent something is never going to be zero. Okay, we can't find any X values that will satisfy this equation. So we actually have no X intercept. Okay. If we look at why intercept? Okay, this is one X zero. So yet Y is equal to E. To the exponent zero 80 X. 10 is going to be one. And so we have the point zero one. Alright, so we know we don't have an X intercept. So we don't cross the X axis and we have a positive Y intercept. So our graph is going to lie In the upper two quadrants because we're gonna draw like this and again we're in rectangular coordinates. So are X axis markings are going to be more spread out than the Y axis like this. Okay, Now we know we have the .01. So we have this right here. Why intercept each of the X. Is an increasing function as X gets bigger and bigger is going to get bigger and bigger exponentially. And our function looks something like this. Okay, that's a rough sketch. So this is the curve, Y is equal to E. To the X. And what we see when we're graphing inequalities like this is that curve is going to split the plane into two. Okay, we have everything kind of above this curve and everything below this curve. And what we want to determine now is for our inequality, which part of that plane is included. Well in this case we have why? Greater than or equal to eat the eggs. That means we want everything above this line. And so everything above this line in our plane is going to be included in our inequality. Now if you get confused with the greater than or less than and you're trying to figure out what part of the plane to include and what not to include. What you can do is pick a particular point on the plane. Okay? So if you pick 00 and we put this into our Inequality on the left hand side, we're going to have zero on the right hand side. We're gonna have E. To the exponent zero which is gonna be one. So we get zero is greater than or equal to one that's not true. Okay So we know that that 10.0 is not included in our inequality. Okay. So we know that that bottom portion below that line, that part of the plane isn't included. Okay. So we know that it's the other point. And again you can pick a point above and test it out and you'll find that it does work. Okay. And so this is going to be the graph of our inequality. And if we go back up to our answer choices, we'll see that we have a graph that looks just like d okay we have the graph we've plotted E. To the X. K. With the Y intercept of one and we've included everything above that curve in our inequality. That's it for this one. Thanks everyone for watching. See you in the next video.

In Exercises 39–45, graph each inequality. y ≤ 2^x

Key Concepts

Inequalities

Exponential Functions

Graphing Techniques

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