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Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 5 - Systems of Equations and InequalitiesProblema 49
Capitolo 6, Problema 49

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 \(\end{cases}\)

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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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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.
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Circles in Standard Form

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.
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Systems of Inequalities

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.
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