Graph the solution set of each system of inequalities.

Question

$y\le x^3-x$

$y>-3$

Accepted Answer

Welcome back everyone. In this problem, we want to graph the solution set for the system of inequalities, Y is less than or equal to two X cubed plus X and Y is greater than negative one. For our answer choices, we have possible solutions for A B. See you and D says there is no solution set. And below these, I've put our own blank graph that can help us to plot and solve for the solution set. Now to find a solution set, we need to graph both of our inequalities and wherever the shaded regions intersect is going to represent our solution set. So let's start by graphing both of our inequalities. Let's start with Y is less than or equal to two XQ plus X. Now first, we'll need our curve here in this case because it's a cubic graph. And then once we have the curve, we can figure out which region or inequality lies in order to shade. So let's first start by solving for our curve or plotting our curve. Let's let Y be equal to two X cubed plus X. Now we'll need a few points here. So let's set up a table of X and Y values. OK? And we're gonna use X and Y values between negative two and positive two. Now, in order to solve we can go ahead and substitute. OK. So let's say for example, that X is negative two, OK? Then at X equals negative two, that means the Y value according to our equation is going to be two multiplied by negative two cubed plus negative two, negative two cubed is 82 multiple, sorry negative two cubed is negative eight multiplied by two is negative 16 plus negative two equals negative 18. OK. So when X is negative two, Y is negative 18, likewise, if we go ahead and substitute values to help us figure out the rest of Y values, then we should get X when X is negative one, Y is negative three, X is zero, Y is zero, X is one, Y is three and when X is two, Y is 18, now that we have these points, we can plot them and connect them to show a smooth curve. OK? So now when we think about it, OK. Let me it's a bit thinner here. So negative two negative 18. OK. It's gonna be down here uh negative one, negative three. It's gonna be somewhere here. 00. OK? Uh 13 and then two positive 18. OK. And we know that our cubic graph is, is gonna look something like this. OK? So it's gonna be something is gonna have a bit of a curve here and then it's gonna continue upward. So now the question is which one of these regions is going to contain our inequalities? It's going to satisfy rather the inequality. Well, let's take a point that's not on our line. So for example, let's take the 0.50. OK. So 50 is over here. So that's our test point no, four or test 0.50. OK, then no, if we put it into our equation testing 50, the X coordinate is five, the Y coordinate is zero. So the question is, is zero less than or equal to two multiplied by five cubed plus five. Well, five cubed is 125 multiplied by two is 250 plus five is 255. So is zero less than equal to 255. Well, it is because zero is less than 255. So that means +50 is in the shaded region for inequality, which means here that if I were to do our shade here, let me just try to stay as close to the curve as I can. And again, our sketch just gives us a rough idea, but we have our answers that we can compare them to. So now here we know that our shaded region is going to be this area in red. OK? Gonna be this entire shaded region now that we have our shaded region for one of our inequalities, let's move on to the other. That is where it says why is greater than negative one? Now, if we think about it, the graph of Y equals negative one, OK is a horizontal line that passes through zero, negative one. In other words, that line is going to be right here. OK? I know because we know that it's greater than negative one. It doesn't include when Y equals negative one. So actually it's going to be a dotted line. OK. Let's put our dots here on our land. And because we are looking for the points where why Y is greater than negative one, that means the shaded region is going to be above the line. OK? Let's put that in blue here. We have our, let me just draw a line here. OK? No, we know that anything above that. OK? Anything above that is going to be or it is going to satisfy the inequality Y is greater than negative one. So now that we have both of our equations, OK? Are both are for inequalities rather what is going to be our solution set? Well, where do our tool graphs interact? I like to think about it as the region where that has all of the colors for our shaded regions. And in this case, you will notice that it's this region right here. OK? So if we were to highlight this region, it's gonna be here with this black line. OK. So our shaded region or sorry, our solution set is going to be this area that I've shaded in black. OK? Uh This should be a straight line down here. Actually, let me just fix this a bit. OK? And now we know that it will look something like that. So this is going to be our shaded region. Let's compare it to our answer choices which one is correct? Well, by comparison, OK, we'll notice that A OK. Is the correct answer? Thus, a represents the correct solution set for our system of inequalities. Thanks a lot for watching everyone. I hope this video helped.

Graph the solution set of each system of inequalities.
$y\le x^3-x$
$y>-3$

Key Concepts

Inequalities

Graphing Inequalities

Systems of Inequalities

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