Problem 82 - College Algebra with Modeling and Visualization

Question

Shade the solutions set to the system. $$ \begin{aligned} &x^{2}+(y+1)^{2} \leq 9\\\ &(x+1)^{2}+y^{2} \leq 9 \end{aligned} $$

Step-by-Step Solution

Answer

Shade the lens-shaped intersection of the two circles.

Step 1: Identify the Inequalities

We have two inequalities to consider: (1) \( x^2 + (y+1)^2 \leq 9 \) and (2) \( (x+1)^2 + y^2 \leq 9 \). Each represents the area inside (and including the boundary of) a circle.

Step 2: Examine the First Circle

The circle from inequality (1) is centered at \( (0, -1) \) with a radius of 3. This is determined by rewriting the inequality as \( (x-0)^2 + (y+1)^2 \leq 3^2 \). The solutions are all points within and on this circle.

Step 3: Examine the Second Circle

The circle from inequality (2) is centered at \( (-1, 0) \) with a radius of 3. This is found by rewriting it as \( (x+1)^2 + (y-0)^2 \leq 3^2 \). Again, the solutions are all points within and on this circle.

Step 4: Find Intersection of the Circles

To find the solution set to the system, we need to determine the overlapping region of the two circles. Graphically, this means shading the region where the two circles intersect.

Step 5: Shade the Intersection Region

First graph the two circles on a coordinate plane. The first circle is centered at \( (0, -1) \) and the second at \( (-1, 0) \). The intersection will generally be a lens-shaped region. This is the solution set since it satisfies both inequalities.

Key Concepts

Algebraic Inequalities

When dealing with algebraic inequalities, we are interested in finding a range or a set of values that satisfy a particular condition expressed in an inequality form. An inequality compares two expressions and represents them using symbols such as \( \leq \) (less than or equal to), \( \geq \) (greater than or equal to), among others.
For example, in the inequalities \( x^2 + (y+1)^2 \leq 9 \) and \( (x+1)^2 + y^2 \leq 9 \), the expressions represent areas that include the boundary in our solutions and ask, "Where can these expressions be less than or equal to 9?"

It's essential to remember:

• "Less than or equal to" includes the edge of the region, not just the inside.
• Inequalities describe a region that corresponds to all the possible solutions.
• When combined, they form a system which can have a common overlapping region or solution set.

Solution Set

The concept of a solution set is foundational in understanding systems of equations and inequalities. A solution set contains all possible solutions that satisfy a given system of inequalities or equations.
In our exercise, we seek the intersection of two inequalities: \( x^2 +(y+1)^2 \leq 9 \) and \( (x+1)^2 + y^2 \leq 9 \). The solution set is the intersecting area between these two circle inequalities.

Keep in mind:

• The solution set reflects the overlap where both conditions are true.
• For visual representation, sketching helps in identifying the overlapping region.
• This set is often a bounded area in geometry, like a shape or region on a graph.

By finding and shading this overlapping area, we solve the problem and visualize the solution set for the entire system.

Circle Equations

Understanding circle equations is a key aspect in graphing and solving inequalities involving circles. A typical circle equation is written in the form: \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) is the center of the circle, and \(r\) is the radius.

For the first circle in our exercise, \( x^2 + (y+1)^2 \leq 9 \), the equation is reformulated to find the center at \((0, -1)\) with a radius of 3.
The second inequality \( (x+1)^2 + y^2 \leq 9 \) translates to a circle centered at \((-1, 0)\) with the same radius.

Some helpful points about circle equations:

• The inequality \( \leq \) signifies all points inside and on the circle, forming a disk.
• Circle properties like center location and radius size help in visualizing the solution set area.
• These are symmetrical shapes that often make double-checking solutions straightforward.

Recognizing these aspects aids in solving systems of inequalities that involve circular regions, making the problem easier to manage and solve.