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}$

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Identify the inequalities given: $x^{2} + y^{2} > 9$ and $x^{2} + y^{2} < 25$.

Recognize that these inequalities represent regions related to circles centered at the origin. The first inequality, $x^{2} + y^{2} > 9$, describes all points outside the circle with radius 3 (since $\sqrt{9} = 3$).

The second inequality, $x^{2} + y^{2} < 25$, describes all points inside the circle with radius 5 (since $\sqrt{25} = 5$).

To find the solution set of the system, look for points that satisfy both inequalities simultaneously. This means points must lie outside the smaller circle (radius 3) but inside the larger circle (radius 5).

Graphically, this solution set is the region between the two circles, excluding the boundaries since the inequalities are strict (greater than and less than, not greater than or equal to or less than or equal to).

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주요 개념

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

Inequalities Involving Circles

Inequalities like x² + y² > r² and x² + y² < R² represent regions outside or inside circles centered at the origin with radii r and R, respectively. Understanding these inequalities helps identify areas on the coordinate plane that satisfy the conditions.

Graphing Solution Sets of Systems of Inequalities

Graphing a system of inequalities involves shading the regions that satisfy each inequality and finding their intersection. The solution set is the overlapping area where all inequalities hold true simultaneously.

Annulus Region Between Two Circles

The system x² + y² > 9 and x² + y² < 25 describes an annulus, the ring-shaped region between two concentric circles with radii 3 and 5. Recognizing this helps visualize and graph the solution as the area between these two circles.

Graph the solution set of each system of inequalities or indicate that the system has no solution.{x2+y2>1x2+y2<16\(\begin{cases}\) x^2 + y^2 > 1 \\ x^2 + y^2 < 16 \(\end{cases}\) {x2+y2>1x2+y2<16