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 - 1)^2 + (y + 1)^2 < 25 \(x - 1)^2 + (y + 1)^2 \geq 16\end{cases}$

Accepted Answer

Hey, everyone here, we are asked to consider the following system of inequalities. The quantity of X minus two squared plus the quantity of Y plus two squared is less than nine. And the quantity of X minus two squared plus the quantity of Y plus two squared is greater than or equal to four. Here we are to graph the solution set or state that there is no solution. Here, we have four answer choice options, answer choice A where we have a larger circle and a smaller circle inside of the larger circle. And the larger circle has a dash line and the smaller circle has a solid line. And lastly, the area in between both of these circles is shaded. Next, we have answer choice B which also has two circles where one smaller circle is inside of the larger circle. And both of these circles have solid lines and the area between both of these circles is shaded. Then we have answer choice C which states we have infinite solutions. And lastly answer choice D which states we have no solution. So the solution set to this given system of inequalities is going to be the area where the shadings from each individual inequality overlap one another. So we first need to graph each inequality. And so we can first start with the first given inequality, which if we recall is the quantity of X minus two squared plus the quantity of Y plus two squared is less than nine. So now looking at this first inequality, we see that it matches the equation of a circle. And so from the quantity of X minus two squared, we see that our circle will translate two units to the right. And then from the quantity of Y plus two squared, we know that the circle will also translate two units down. And then the value on the right side of the inequality tells us our radius. So we know that nine is the same as three squared which tells us that our radius for this circle is going to be three. So now with these components of our circle identified, we can go ahead and graph our inequality. So we see we end up with a circle again translated two units to the right and two units down with a radius of three. But more importantly, we see that the outline of this circle, the line of the circle is dashed. And this is due to the inequality being a strict inequality symbol. And then we also see that this expression on the left side of the inequality is less than nine or less than the radius. Which is why the area in between the circle or inside of the circle is shaded because again, this expression is less than the radius. So now that we have graphed our first inequality, we can move on to graphing the second inequality where we have the quantity of X minus two squared plus the quantity of Y plus two squared is greater than or equal to four. Now we see with this second inequality that this second circle shares some of the same properties as our first circle. So first, we see that again, our circle translates two units to the right and two units down. But our radius here is different here we have four. So four is the same as two squared which tells us our radius for the second circle or the second inequality is going to be two. So now we just want to graph this in equality and we see that we do in fact have a circle translated two units to the right two units down with a radius of two. But here we have a solid line for the outline of our circle. And this is due to the greater than or equal to symbol the non strict inequality symbol. And we also see that the area outside of the circle is shaded. And this is because that this expression is greater than the radius. So now with both of our inequalities graph, we can just combine the two graphs to find our solution set or the solution graph here. And so now combining, we see we have two circles, one circle inside of the other. And our smaller circle is still that solid line and the larger circle is that dashed line. And now combining these shaded regions, we see that we end up with the area in between these circles as part of our solution. So the other thing we need to know is that the points on the dash line are not included because again, the dash line means we do not include those values since we had less than nine from our first inequality. But we do include the points on the solid line as well as the area in between the two circles. So now with our combined graph, we can look back at our solutions and we see here that we are left with answer choice. A 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.
$\begin{cases}(x - 1)^2 + (y + 1)^2 < 25 \\(x - 1)^2 + (y + 1)^2 \geq 16\end{cases}$

Key Concepts

Graphing Circles from Inequalities

Systems of Inequalities

Inequalities with 'Greater Than or Equal To' and 'Less Than' Signs

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