Problem 36 - College Algebra and Calculus: An Applied Approach

Question

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}x^{2}+y^{2} \leq 25 \\ x^{2}+y^{2} \geq 9\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system is the ring-shaped area between the circle with radius 5 and the circle with radius 3, both center at the origin, which includes both the circles themselves.

Step 1: Identify Equations

The equations represents two circles. The first equation \(x^{2}+y^{2} \leq 25\) represents a circle centered at the origin (0,0) with radius 5. The inequality means all the points inside and on the circle are included in the solution set. Similarly, the second equation \(x^{2}+y^{2} \geq 9\) is another circle centered at the origin with radius 3. However, in this case the inequality tells us that all points outside and on the circle are included.

Step 2: Draw Circles

Draw the first circle with center as the origin (0, 0) and 5 as the radius. Then draw the second circle with the origin as the center and 3 as the radius.

Step 3: Find Area of Interest

According to the inequalities, the solution is for points that are inside or on the circle with radius 5 (including the circle itself) and outside or on the circle with radius 3 (including the circle itself). Therefore, the area of interest for the system of inequalities is the ring-shaped area between these two circles.

Key Concepts

Circle Equations

Understanding the equation of a circle is fundamental in graphing systems of inequalities when circles are involved. The general form of a circle equation is \(x - h)^2 + (y - k)^2 = r^2\), where \(h, k\) is the center of the circle and \(r\) is the radius.

In the given exercise, we have two inequalities involving circle equations. The first \(x^2 + y^2 \leq 25\) represents a circle centered at the origin \(0, 0\) with a radius of 5 units. That is, every point \(x, y\) that satisfies the inequality lies inside or on a circle that has a 5-unit radius from the center. Additionally, the second inequality \(x^2 + y^2 \geq 9\) defines another circle, also centered at the origin, with a radius of 3 units. This encompasses all points outside or on the circle. These equations form the basis of the graphical solution set for this problem.

To improve comprehension, visualize the circle as a round fence and the inequalities \(\leq\) and \(\geq\) indicate which side of the fence (inside for \(\leq\), outside for \(\geq\)) the solution lies. By sketching these circles on a graph, students can better conceptualize the areas represented by the inequalities.

Inequality Solutions

Inequality solutions are often more complex than simple equations because they represent a range of values rather than a single solution. For instance, a linear inequality like \(y > x\) is not satisfied by a single point, but rather by an entire region in the coordinate system.

In the context of the given circle equations, we are dealing with a system of inequalities, which means we're looking for a set of points that satisfy both conditions simultaneously. This requires us to determine where the regions represented by each inequality overlap. The inequality \(x^2 + y^2 \leq 25\) includes all points inside or on the circle with a radius of 5, while \(x^2 + y^2 \geq 9\) includes all points outside or on the circle with a radius of 3.

To find the solution to the system, one must locate the region that satisfies both inequalities at once, which, in this case, is the 'ring' or 'annulus' formed between the two circles. Students can think of this as seeking the common ground in a Venn diagram, where each inequality is a set, and the solution is their intersection.

Graphical Representation of Inequalities

Graphically representing inequalities can help students visualize the solution sets in a more tangible way. When graphing inequalities, especially those representing regions like circles or half-planes, it's common to use shading to indicate where the solutions lie.

In our exercise, each circle equation corresponds to a boundary of the solution set. The first inequality, \(x^2 + y^2 \leq 25\), suggests we shade the interior of the circle defined by a radius of 5, as it includes all the points equal to or less than that distance from the center. Conversely, for the inequality \(x^2 + y^2 \geq 9\), the shading would occur outside the circle of radius 3 to include all points at or beyond that distance from the center.

The graphical solution for the system is the area where both shadings overlap, which is visually represented by a different pattern or color. This 'shared' area between the two circles is the solution set for the system of inequalities. When students draw these on a graph themselves, they should use different shading patterns to clearly distinguish between the regions contributed by each inequality to accurately identify the solution set.