Problem 75 - Algebra and Trigonometry

Question

Determine if each system has no solution or infinitely many solutions. $$\left\\{\begin{array}{l} {(x+4)^{2}+(y-3)^{2} \leq 9} \\ {(x+4)^{2}+(y-3)^{2} \geq 9} \end{array}\right.$$

Step-by-Step Solution

Answer

The given system of inequalities has infinitely many solutions.

Step 1: Analyze the inequalities

The given inequalities are essentially equations of circles in the coordinate plane. The circle is same for both cases since the expressions on left are same in both inequalities. The center of the circle is located at (-4,3) and the radius, r, equals to the square root of 9, which is 3.

Step 2: Interpret the inequalities

The inequality \( (x+4)^{2}+(y-3)^{2} \leq 9 \) implies the set of points (x, y) that are located inside or on the boundary of the circle, while the inequality \( (x+4)^{2}+(y-3)^{2} \geq 9 \) represents the set of points (x, y) that are outside or on the boundary of the circle.

Step 3: Find the common solution

For a point to satisfy both inequalities, it would need to be at the same time inside or on the boundary of the circle and outside or on the boundary of the same circle. Clearly, there is a contradiction when trying to be inside and outside at the same time. Therefore, the only remaining points that could satisfy both conditions are those on the boundary of the circle, where both inequalities meet. Meaning, all points where \( (x+4)^{2}+(y-3)^{2} = 9 \)

Step 4: Determine the conclusion

Drawing a conclusion from the previous step, a common solution to these inequalities exists. Precisely, there are infinitely many solutions, because all points on the boundary of the circle satisfy both inequalities. These points represent the simultaneous solution of the system. This means the answer is infinitely many solutions.

Key Concepts

Systems of Inequalities

In the world of mathematics, systems of inequalities involve more than one inequality in a set. These inequalities share common variables and must be considered together, as a system. To find solutions, we seek to identify the values that satisfy all inequalities simultaneously. Let's explore our specific example:

• First Inequality: \((x+4)^{2}+(y-3)^{2} \leq 9\), which includes points inside or exactly at the boundary of a circle.
• Second Inequality: \((x+4)^{2}+(y-3)^{2} \geq 9\), which comprises points outside the circle or exactly at the boundary.

To solve a system like this, we aim to find a set of points that will satisfy both conditions simultaneously. However, with one inequality suggesting points inside or at the boundary and the other suggesting outside or at the boundary, the only common ground is the boundary itself.

Coordinate Plane

Visualizing inequalities or equations involving circles on a coordinate plane can simplify understanding. The coordinate plane is like a map with two axes, the x-axis and the y-axis. They intersect at the point (0,0), known as the origin. Each point on the plane is identified by an ordered pair \((x, y)\). When interpreting our system of inequalities:

• The expression \((x+4)^{2}+(y-3)^{2}\) determines a set of points centered at (-4, 3).
• The plane serves as a tool to visualize how inequalities define different regions. Here, these involve a central circle.

By placing equations and inequalities on this plane, we can better see where their solution sets, if any, intersect or diverge. In our case, the center (-4, 3) of the circle provides a focal point for constructing the boundary circle which representations help us grasp the concept of being inside, outside, or right on the boundary.

Circle Boundary

When dealing with circle equations in inequalities, understanding the circle's boundary is critical. It acts as a divider between points inside, outside, and right on the circle.

Defining the Circle

A circle's equation built on a coordinate plane has a general form: \((x-h)^{2} + (y-k)^{2} = r^{2}\). Here,

• \((h, k)\) represents the circle's center.
• \(r\) is the radius

For our problem,

• Center: \((-4, 3)\)
• Radius: 3

Inequality Implications

• The boundary equation \((x+4)^{2}+(y-3)^{2} = 9\) finds points exactly at the circle's edge.
• Points where \((x+4)^{2}+(y-3)^{2} < 9\) lie within the circle.
• Points where \((x+4)^{2}+(y-3)^{2} > 9\) lie outside.

Understanding these implications helps us solve systems like ours, as only points exactly on this boundary could possibly meet both given conditions.

Solution Sets

Solution sets in equations and inequalities are the collections of values that satisfy a given system. For our exercise, the primary task was to determine if the inequalities have no solution or infinitely many solutions.

Interpreting the Result

Based on our system of inequalities:

• The first inequality represents all points on or inside the circle.
• The second inequality represents all points on or outside the circle.
• The intersection of these two sets is precisely the circle's boundary itself.

Conclusion

Since the boundary is a continuous set of points forming a circle, there are infinitely many solutions residing on it. This is because every point on the circle's edge satisfies both the inside and the outside conditions of the inequalities. Thus, we find that the solution set is indeed infinite but constrained to the circle's boundary.