Graph the solution set of each system of inequalities.

Question

$y\ge3^{x}$

$y\ge2$

Accepted Answer

Welcome back everyone. In this problem. A system of inequalities is given as Y is less than or equal to negative two to the power of X and Y is less than or equal to negative three. We want to graph its solution set. A shows possible solutions B shows another possible solution C shows a third possible solution. And D says there is no solution set. I know below our answer choices. I've pasted a graph, OK. A blank graph that we can use to plot our inequalities. Now, if we're going to graph the solution set, we need to graph both of our inequalities on our grid. And then wherever the regions intersect is where the solution set will be. So let's start by trying to graph both of our inequalities. Let's start OK. Let's start with Y is less than or equal to negative two to the power of X. Now, if we are going to figure out where this region is first, we'll need to draw the line that represents Y equals negative two X. OK. So if we let Y equals negative two to the power of X, sorry, my apologies. I said it incorrectly. A while ago. But if we let Y equal negative two to the power of X because we know that this will be, this will make a curve. OK, then we can draw up a table of X and Y values. OK? So let's say that our X and Y values are between negative three and one. So that's negative three, negative two, negative 10 and one. OK? Now what are, or why values going to be well to figure them out, we can substitute. In other words, at X equals negative three, Y is going to be equal to negative two. OK? To the power of negative three, I know we know two to the power of negative three equals one divided by two to the third which is an eighth. So therefore, Y is going to be equal to the negative value of an eighth. Therefore, when X is negative three, Y is negative 18. Likewise, when X is negative two, Y is negative a quarter when X is negative one, Y is negative a half when it's a zero, Y is negative one and when it's one, Y is negative two. So now we have enough values to plot our Asymptote. So when we go ahead and do that, OK. So we have, when X is negative three Y is negative and eight. OK? So let's put that here when X is negative uh when X is negative two, OK? Why is negative a quarter? OK. Oops should be on the other side. My apologies. So Y is negative a quarter when X is negative one, Y is negative a half when X is zero, Y is negative one and when X is one, Y is negative two. OK? So we know that it's gonna be an Asymptote that's gonna look something like this. OK? Oops, let me draw that, try to draw that a bit better. But I think, I think we have an idea of how our graph will look. OK. So it will look something like that curve. No know that we have our curve. OK. Then let's try to draw our next graph. OK? Or our next inequality. But before we do that very important, we need to share the region, we know that Y is less than or equal to negative two to the power of X. So that means it's going to include all of the points on the curve and the shaded region will be below our curve. So that means our shaded region here, if we draw this to fit our curve, OK, then our shaded region is going to be below our curve in this area. So it's going to look something like this. OK. So this is going to be our shaded region. So now that we've done that we can put our next inequality OK? And now we know that for our next inequality, it says that Y is less than or equal to negative three. No, the graph of Y equals negative three is a horizontal line that passes through zero, negative three. In other words, this is going to be our line for Y is equal to negative three. And if Y is less than or equal to negative three, that means the shaded region will be below this line which is going to be a solid line because it includes all of the points where Y is equal to negative three. So now if we go ahead and draw that, OK, if we go ahead and put that here, then we know, let me just fix that a bit. Yes. So we know then that our shaded region is going to be below it, which is down here. So this is going to be the shaded, this is this shaded region represents the inequality. Why is less than equal to negative three? Now, where's the solution system? Well, as we said, it's the intersection of the inequalities. The way I like to think about it is that it is the area where all of our colors are shaded, the area that has all of the colors that we have already put in. I know if you notice here that region is going to be this area that I'm about to highlight in our black line. OK. So our shaded region OK is going to be in between this area. OK. So this is going to be the region for our solution set the region where it satisfies both inequalities. Now, when we compare this region to the rest of our answer choices, this should tell us that C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Graph the solution set of each system of inequalities.
$y\ge3^{x}$
$y\ge2$

Key Concepts

Inequalities

Graphing Exponential Functions

Shading Regions in Graphs

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