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

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

문제 51

6장, 문제 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}$

검증된 단계별 안내

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).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.

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

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

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}$