Graph the solution set of each system of inequalities or indic...

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution.

$\begin{cases}x^2 + y^2 \leq 16 \x + y > 2\end{cases}$

Accepted Answer

Hey, everyone here, we are asked to consider the following system of inequalities X squared plus Y squared is less than or equal to 36 X plus Y is greater than six. Here we are asked to graph the solution set or that there is no solution. Here, there are four answer choice options. First answer choice A which has a solid line circle and also a solid line that passes through the circle. And the shaded area is above the line but also enclosed by the circle. Then we have answer choice B with the same line and solid line circle. But the line that passes through this circle is now dashed. And then the area below the line that is still enclosed by the circle is shaded. Then we have answer choice C which is another graph of the same circle and line. The circle is a solid line circle and the line is dashed and the shaded region is the area above the line that is enclosed by the circle. Lastly, we have answer choice D which just states that we have no solution. So here the solution set to this system of inequalities will be the area where the shadings from each inequality overlap one another. So that means to solve this problem, we first need to graph each individual inequality. So we can start with our first given inequality which if we recall is X squared plus Y squared is less than or equal to 36. So here we see that this inequality matches the equation of a circle that is centered at the origin with a radius of six. So now graphing, we see that we do in fact have a circle centered at the origin with the radius of six. And we also see that the shaded region is inside the circle itself because X squared and Y squared cannot be more than the radius. And we also see that the line of the circle, the points on the line are included because the, because X squared and Y squared can be equal to 36. And here we see on this graph that the extreme values are included. So now that we have graphed, our first inequality, we can move on to graphing the second given inequality which is three X plus Y is greater than six. So here we see that we have an equation of some line. So graphing this inequality, we get a dash line and include it are the points that of the Y intercept and the X intercept just to show or indicate that the graph supposedly passes through these points. And we see that the line is a broken dash line because the points in the line itself are not included due to our greater than symbol. So now with both of our graphs, our individual graphs for each inequality graph here, we can now combine these graphs to get our solution set. So combining, we see that we have our solid line circle with the area inside shaded and we also have our dash line that passes through our circle and then the area above this line is shaded. So our solution set is going to be the area areas that overlap. So we have the area above the line that is still enclosed by the circle. And we need to notice that the points on the line are not included due to the dash line, but the points on the circle are included because it is a solid line. So now with our combined grant, we see that we are left here with answer choice. C thanks for watching. I hope you found this video helpful and I'll see you again next time.

Graph the solution set of each system of inequalities or indicate that the system has no solution.
$\begin{cases}x^2 + y^2 \leq 16 \\x + y > 2\end{cases}$

Key Concepts

Graphing Circles and Inequalities

Graphing Linear Inequalities

Solution Set of a System of Inequalities

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