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

Ch. 5 - Systems of Equations and Inequalities

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

Problema 51

Capitolo 6, Problema 51

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}$

Guida verificata passo dopo passo

Identify the inequalities as representing circles centered at (2, -2). The first inequality $\left(x - 2\right)^2 + \left(y + 2\right)^2 < 9$ 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).

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Graphing Circles from Inequalities

Inequalities involving expressions like (x - h)² + (y - k)² < r² represent the interior of a circle centered at (h, k) with radius r. The inequality < indicates points inside the circle, while ≤ includes the boundary. Understanding how to graph these regions is essential for visualizing solution sets.

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Circles in Standard Form

Systems of Inequalities

A system of inequalities requires finding the set of points that satisfy all inequalities simultaneously. This often involves identifying overlapping regions on the graph. Recognizing how to combine solution sets helps determine the feasible region or if no solution exists.

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Systems of Inequalities

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

Inequalities with ≥ or > define regions outside or on the boundary of a circle, while < or ≤ define regions inside or on the boundary. Distinguishing between strict and inclusive inequalities affects whether the boundary circle is part of the solution set, influencing the graph's shading and boundary style.

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Linear Inequalities

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In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. x²+y²≤1, y−x²>0

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Find the partial fraction decomposition for 1/x(x+1) and use the result to find the following sum:

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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

Risposta video verificata per un problema simile:

Concetti chiave

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