Problem 50 - College Algebra

Question

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x^{2}+y^{2}>1 \\ x^{2}+y^{2}<9 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution set area lies in an annular region between the circle with radius 1 and the circle with radius 3, not inclusive of the circles themselves.

Step 1: Understand inequalities

Both \(x^2 + y^2 > 1\) and \(x^2 + y^2 < 9\) are equations of a circle. The former is a circle with a radius of 1 unit at the origin (0,0), and the inequality sign '>1' denotes points lying outside the circle. The latter is a circle with a radius of 3 units at the origin, and '<9' denotes points lying inside this circle.

Step 2: Identify potential solution area

The potential solution area for both these inequalities combined is an annular region between these two circles (outer radius 3 units, inner radius 1 unit). This region includes all points that are farther than 1 unit from the origin and closer than 3 units to the origin.

Step 3: Graph the inequalities

On the graph, plot the two circles. The area of the solution between them is delimited, but the circles themselves are not part of the solution (because the inequalities do not contain 'or equals' notation). Therefore, represent this using a dashed line for the circles and shading the area of the solution between the two circles.

Key Concepts

Inequalities

Inequalities express relationships where one quantity is greater than or less than another. In mathematics, we often see this with inequality signs such as ">", "<", ">=", and "<=". It allows us to show a range of values that satisfy certain conditions rather than pinpointing an exact number. Inequalities are very practical when dealing with ranges, like temperatures, distances, or financial limits.

For the problem at hand, the inequalities are based on circles:

• The first inequality, \(x^2 + y^2 > 1\), implicates areas of the plane where points lie strictly outside the circle. The boundary (the circle itself) is not included because of the strict greater-than sign ( > ).
• The second inequality, \(x^2 + y^2 < 9\), represents points within a larger circle where the radius is 3. This is an open circle, excluding the perimeter as well ( < ).

Understanding these inequalities tells us which areas are to be included or excluded, guiding us to correctly graph the solution.

Graphing Circles

Graphing circles involves plotting all points that are equidistant from a central point. This distance is the radius of the circle, and the equation for a circle centered at the origin is \( x^2 + y^2 = r^2 \), where \(r\) is the radius.

In our exercise, we have two circles:

• The first circle has a radius of 1, described by the equation \(x^2 + y^2 = 1\). Because the inequality is \(x^2 + y^2 > 1\), we need points outside this circle.
• The second circle has a radius of 3, described by \(x^2 + y^2 = 9\). Here, \(x^2 + y^2 < 9\) means points inside this circle.

When graphing, use dashed lines to show these circles are not included (since we do not have ">=" or "<=" signs). On the graph, distinctly shade the area indicating where points satisfy both inequalities.

Solution Set

The solution set of a system of inequalities is the region that satisfies all conditions within those inequalities. When dealing with two or more inequalities, like in this exercise, it’s essential to identify overlapping areas that meet every inequality’s requirement.

For this problem, the solution is an annular region (a ring-shaped area) between two circles centered at the origin:

• Points are outside the inner circle with radius 1.
• Points are inside the outer circle with radius 3.

These combined conditions precisely frame a ring between the two boundaries, but not touching either circle. On a graph, this region would be clearly shaded, emphasizing it is the only area where the combined inequalities hold true. Such exercises help sharpen skills in visualizing mathematical conditions and interpreting complex constraints.