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.

$\begin{cases}x^2 + y^2 > 1 \x^2 + y^2 < 16\end{cases}$

Accepted Answer

Hey, everyone here we are asked to consider the following system of inequalities X squared plus Y squared is greater than nine and X squared plus Y squared is less than 25. Here we are asked to graph the solution set or state that there is no solution. Here we have four answer choice options, answer choice A which displays a graph of a circle that is a dashed line circle in the area inside of the circle is shaded. Then we have answer choice B with a graph displaying two circles, one larger circle with a smaller circle inside of it. And then both of these circles are dashed lines and the area in between both of these circles is shaded. Then we have answer choice C which states that we have infinite solutions. And lastly answer choice D which states we have no solution. So here in this problem, the solution set it to this system of inequalities is going to 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 begin with our first given inequality which if we recall is just X squared plus Y squared is greater than nine. Now, with this inequality, we see that it matches the equation of a circle that is centered at the origin. And we see from the value on the right side of our inequality that our radius is going to be three. So now graphing our inequality, we see that we do in fact have a circle centered at the origin with a radius of three. And that the one circle is a dash line because of our strict inequality symbol. And then the area outside of the circle is shaded because we have X squared plus Y squared set greater than the radius. So we have graphed our first inequality onto graphing our second inequality which is again, X squared plus Y squared is less than 25. And so here again, we see that we have an equation of another circle that is centered at the origin, but now with a radius of five. So just graphing this inequality, we get a circle centered at the origin with a radius of five. And again, we have another dashed line circle because of our strict inequality symbol. And because we have X squared plus Y squared set less than the radius, we have the area inside of the circle shaded. And now that we have graphed both of our inequalities individually to find our solution set, we just have to combine both of these graphs So combining, we get a graph with two circles, both circles have dash lines and we have the larger circle enclosing the smaller circle that are both centered at the origin. And then we see that after combining the shaded regions that we are left with the shaded region being in between both of our circles. And so it is important to note that all of the points lying on both of our circles are not included in the solution because we have dashed lines. So the only part of the solution of our graph here is this shaded region. So now with our graph, we can look at our possible solutions here and we see that we are left with answer choice B 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.
$\begin{cases}x^2 + y^2 > 1 \\x^2 + y^2 < 16\end{cases}$

Key Concepts

Inequalities Involving Circles

Graphing Solution Sets of Systems of Inequalities

Annulus Region Between Two Circles

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