Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. ﻿{(x−1)2+(y+1)2\u003c25(x−1)2+(y+1)2≥16\begin{cases}
(x - $$1)^2 + (y$$ + $$1)^2$$ \u003c 25 \
(x - $$1)^2 + (y$$ + $$1)^2$$ \geq 16
\end{cases}
{(x−1)2+(y+1)2\u003c25(x−1)2+(y+1)2≥16​

Accepted Answer

Identify the inequalities as representing circles centered at (2, -2). The first inequality $$\left(x - 2\$$right)^2$$ + \left(y + 2\$$right)^2$$ \u003c 9$$ describes all points inside a circle with radius 3 (since $$\sqrt{9} = 3). $$The second inequality $$\left(x - 2\$$right)^2$$ + \left(y + 2\$$right)^2$$ \geq 4$$ describes all points outside or on the boundary of a circle with radius 2 (since $$\sqrt{4} = 2). To $$find the solution set of the system, look for points that satisfy both inequalities simultaneously. This means points must lie inside the larger circle (radius 3) but outside or on the smaller circle (radius 2). Graphically, this solution set is the region between the two circles, including the boundary of the smaller circle but excluding the boundary of the larger circle. To sketch the solution, draw two concentric circles centered at (2, -2) with radii 2 and 3. Shade the area between them, making sure the inner circle's boundary is included (solid line) and the outer circle's boundary is excluded (dashed line).

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}$

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Key Concepts

Graphing Circles from Inequalities

Systems of Inequalities

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

Graph the solution set of each system of inequalities or indicate that the system has no solution. {(x−1)2+(y+1)2<25(x−1)2+(y+1)2≥16\(\begin{cases}\)(x - 1)^2 + (y + 1)^2 < 25 \\(x - 1)^2 + (y + 1)^2 \(\geq\) 16\(\end{cases}\){(x−1)2+(y+1)2<25(x−1)2+(y+1)2≥16