Problem 35 - College Algebra and Calculus: An Applied Approach

Question

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}x^{2}+y^{2} \leq 16 \\\ x^{2}+y^{2}<1\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the given system of inequalities is the region between the two circles, including the boundary of the larger circle but excluding the boundary of the smaller one.

Step 1: Interpret the Inequalities

Consider the given inequalities: \(x^2 + y^2 \leq 16\) and \(x^2 + y^2 < 1\). These represent circles centered at the origin (0,0). The radius of the first circle is 4 and the second circle is 1. Also note that the first inequality includes the boundary (the circle itself) and the inner area, while the second inequality only includes the area inside the circle.

Step 2: Draw the Circles

Proceed by sketching the circles on a coordinate plane. For \(x^2 + y^2 \leq 16\), draw a circle with center at (0,0) and radius of 4. Make the circle a solid line to indicate that the points on the circle are included in the solutions. For \(x^2 + y^2 < 1\), draw a smaller circle within the first one having the same center (0,0) and a radius of 1. This time, draw a dashed or dotted line because the points on this circle are not included in solution.

Step 3: Identify the Solution Set

Observe that the common area (intersection) of the two inequalities is the area between the two circles. Since the first inequality (\(x^2 + y^2 \leq 16\)) includes the points on the larger circle and inside it, and the second inequality (\(x^2 + y^2 < 1\)) does not include the points of the smaller circle, only those within it; therefore, the solution is an annular region (a ring-shaped area) lying between the two circles, including the boundary of the larger circle but not the smaller one.

Key Concepts

Graphing Inequalities

Graphing inequalities involving circles can be visualized as shading areas on a coordinate plane. When dealing with inequalities like \(x^2 + y^2 \leq 16\), we are working with a solid circle because the inequality includes the boundary of the circle itself.
To graph the inequality \(x^2 + y^2 \leq 16\), we draw the circle with a solid line to show all points on or inside the circle are part of the solution. For \(x^2 + y^2 < 1\), we use a dashed line circle to show that points on the boundary are not included.
The key is understanding the types of lines:

• Solid Line: Includes the points on the boundary circle (\(\leq\) or \(\geq\)).
• Dashed Line: Excludes the boundary points (\(<\) or \(>\)).

By shading the appropriately defined areas, one can fully visualize the solution set.

Solution Set

The solution set is the overlap or the intersection of the areas that satisfy each inequality in the system. In this exercise, both inequalities represent circles centered at the origin, but with different radii. The solution set is the space lying between the larger circle and the smaller one.
The larger circle \(x^2 + y^2 \leq 16\) captures an area where points within or on the circle satisfy the inequality. The smaller circle \(x^2 + y^2 < 1\) limits this region further by excluding points on its boundary and only accepting those inside.
Hence, the solution is an annular region:

• All points outside and up to the larger boundary.
• Excluding points on or inside the smaller boundary.

This forms what looks like a donut shape, visually emphasizing the area between these two circles.

Algebraic Inequalities

Algebraic inequalities describe a range of potential solutions on a coordinate plane. They expand ways we express relationships between algebraic expressions and geometrical shapes. Inequalities like \(x^2 + y^2 \leq r^2\) form circles with radius \(r\).
When faced with these kinds of problems, it's crucial to identify:

• The center of the circle, usually given by the coordinates \((h, k)\).
• The radius \(r\), that dictates the size of the circle.

By translating these into a physical region on the graph, algebraic inequalities help define specific areas. Whether the area includes boundary points or not depends on whether the inequality uses \(\leq\), \(<\), \(\geq\), or \(>\). Understanding these symbols helps delineate exactly where a solution area lies.

Circle Equations

Circle equations represented as \(x^2 + y^2 = r^2\) directly correlate with geometric circles on a graph. These equations allow us to translate algebraic expression into a visual form.
For circle equations:

• Standard Form: \((x - h)^2 + (y - k)^2 = r^2\).
• Center: Given by \((h, k)\), usually the origin \((0,0)\) if simplified.
• Radius: Equals \(r\), the square root of the constant term on the right side.

In inequalities, this form helps set the groundwork for graphing. The difference between equality and inequality in these equations decides whether the circular boundary is included or not. Thus, circle equations are fundamental to understanding geometric representations of algebraic conditions.