Skip to main content
Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 5 - Systems of Equations and InequalitiesProblem 49
Chapter 6, Problem 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\(\end{cases}\)

Verified step by step guidance
1
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).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
Recommended video:
5:18
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.
Recommended video:
Guided course
6:19
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.
Related Practice
Textbook Question

In Exercises 47–52, solve each system by the method of your choice. {2x2+xy=6x2+2xy=0\(\begin{cases}\)2x^2 + xy = 6 \(\x\)^2 + 2xy = 0\(\end{cases}\)

1
views
Textbook Question

In Exercises 49–50, solve each system for x and y, expressing either value in terms of a or b, if necessary. Assume that a ≠ 0, b ≠ 0 5ax + 4y = 17 ax + 7y = 22

Textbook Question

In Exercises 47–48, solve each system by the method of your choice. (x + 2)/2 - (y + 4)/3 = 3 (x + y)/5 = (x - y)/2 - 5/2

Textbook Question

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

3
views
Textbook Question

In Exercises 47–48, solve each system by the method of your choice. (x - y)/3 = (x + y)/2 - 1/2 (x + 2)/2 - 4 = (y + 4)/3

1
views
Textbook Question

In Exercises 47–52, solve each system by the method of your choice. {−4x+y=12y=x3+3x2\(\begin{cases}\)-4x + y = 12 \(\y\) = x^3 + 3x^2\(\end{cases}\)