Problem 75 - College Algebra

Question

Without graphing, in Exercises 73–76, 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 which are made up of points that lie on the boundary of the circles represented by these inequalities.

Step 1: Identify the inequalities

The system contains two inequalities, each representing a circle with the same center, but differing radii: \[\ (x+4)^{2}+(y-3)^{2} \leq 9 \] and \[(x+4)^{2}+(y-3)^{2} \geq 9 \]

Step 2: Relate the inequalities

The first inequality represents a circle, including its interior, with radius 3 or less. The second inequality represents a circle with radius of 3 or more, including everything outside the circle. Since both circles have the same center, the only possible solution for this system would be the points that are exactly on the boundary of both circles.

Step 3: Identify if the system has a solution

Looking at these inequalities, we can see that all points that lie on the boundary of these two circles (meaning \( (x+4)^{2}+(y-3)^{2} = 9 \)) will satisfy both inequalities. Therefore, the system of inequalities has infinitely many solutions.

Key Concepts

Understanding Circles in the Context of Inequalities

In a system of inequalities like the one given in the exercise, circles play a crucial role. A circle's equation is typically written in the standard form \[(x-h)^2 + (y-k)^2 = r^2\] where

• \((h, k)\) is the center of the circle.
• \(r\) is the radius of the circle.

In the given system, both inequalities \[(x+4)^2 + (y-3)^2 \leq 9\] and \[(x+4)^2 + (y-3)^2 \geq 9\]represent circles with the same center at \((-4, 3)\) and a radius of 3 (since \(9\) is the square of \(3\)). The first inequality (\(\leq 9\)) includes the region inside and on the boundary of the circle, while the second inequality (\(\geq 9\)) includes the region outside and on the boundary of the circle. By understanding these regions, you're identifying the set of points relative to the location and boundary of the circle.

Exploring Inequalities and Their Meanings

Inequalities are expressions that utilize symbols such as \(<, >, \leq,\) and \(\geq\) to define the relationship between quantities. They describe ranges of values rather than precise amounts. When dealing with inequalities and circles, the expressions \(\leq 9\) and \(\geq 9\) indicate areas relative to the circle's circumference.

• The inequality \((x+4)^2 + (y-3)^2 \leq 9\) encompasses an area including the circle's interior.
• The inequality \((x+4)^2 + (y-3)^2 \geq 9\) encompasses the area outside the circle (and the circle's boundary).

These expressions guide you in determining which side of a circle's boundary points can exist. Here, they restrict solutions to the boundary, as it is the only consistent region satisfying both conditions.

Infinitely Many Solutions in Systems of Inequalities

In the context of this exercise, determining whether a system has infinitely many solutions involves investigating the intersection set of all solutions to the inequalities. Here, two inequalities demarcate regions on a coordinate plane.The given system, \[\begin{align*} (x+4)^{2}+(y-3)^{2} \leq 9 \quad \text{and} \quad (x+4)^{2}+(y-3)^{2} \geq 9 \end{align*}\] focuses all solutions around the exact circle defined by equation \((x+4)^2 + (y-3)^2 = 9\). This is the boundary of both circles, encompassing every point where their conditions intersect.
The boundary, symbolizing the circumference of the circle, has infinitely many solutions since it consists of uncountable points lying on this circle. Here, every point on this circumference satisfies both inequalities perfectly, resulting in the concept of infinitely many solutions. Therefore, comprehending this aspect aids in visualizing how points align with geometric shapes while satisfying algebraic conditions.