Problem 76 - College Algebra

Question

Without graphing, in Exercises 73–76, determine if each system has no solution or infinitely many solutions. $$\left\\{\begin{array}{l} (x-4)^{2}+(y+3)^{2} \leq 24 \\ (x-4)^{2}+(y+3)^{2} \geq 24 \end{array}\right.$$

Step-by-Step Solution

Answer

This system has infinitely many solutions.

Step 1: Interpret the inequalities

These inequalities represent two circles centered at (4, -3). The first inequality \((x-4)^{2}+(y+3)^{2} \leq 24\) represents a circle with radius \(\sqrt{24} = 4.9\) and including points within the circle. The second inequality \((x-4)^{2}+(y+3)^{2} \geq 24\) represents a circle with the same radius and including points on the circle and outside of it.

Step 2: Analyze the system

In our context, a solution of the system of inequalities is a point in the plane which is true for both inequalities. Since the second inequality include points within the circle and also points that lie outside, and the first one only points inside or on the circle, the 'overlap' area of both circles (the area that satisfy both inequalities) is the first circle. Thus clearly it has infinitely many solutions.

Key Concepts

Circle

When dealing with systems of inequalities that involve circles, it is essential to understand the basic equation of a circle in coordinate geometry. A circle can be described with the equation: \[(x - h)^2 + (y - k)^2 = r^2\] Here,

• \(h\) and \(k\) are the coordinates of the circle's center.
• \(r\) is the radius of the circle.

The circle in this problem is centered at (4, -3) with a radius of \( \sqrt{24} \approx 4.9 \). The inequality \((x-4)^{2}+(y+3)^{2} \leq 24\) represents the circle and all the points inside or on it, whereas \((x-4)^{2}+(y+3)^{2} \geq 24\) represents the circle and points outside of it.Understanding how these inequalities modify the circle is key to figuring out solutions, as they define different regions on the plane: within, on, or beyond the circle's boundary.

Inequalities

Inequalities define regions on a graph. They are pivotal in indicating which areas solutions can occupy. In this case, the inequalities involve a circle and are expressed as\(\leq\) and \(\geq\):

• \((x - 4)^2 + (y + 3)^2 \leq 24\) — Points that satisfy this inequality lie inside or on the circle.
• \((x - 4)^2 + (y + 3)^2 \geq 24\) — Points that satisfy this inequality lie on or outside the circle.

Inequalities set constraints. Thus, by determining the overlapping regions that satisfy both inequalities, we can find solutions in the provided system. The first inequality includes all points within the circle's boundary, and the second covers the region beyond that boundary.

Solutions of a System

Finding solutions to a system of inequalities requires identifying points that satisfy all given conditions. In this problem, we have inequalities that define a circle and different regions related to it. The first inequality (\((x-4)^2 + (y+3)^2 \leq 24\)) implies that we are looking at the set of all points within this circle, effectively including the boundary. Next, the second inequality (\((x-4)^2 + (y+3)^2 \geq 24\)) means we are considering points on the circle and outwards. Ultimately, our task is to find where these conditions both submit truth simultaneously. Here, all points satisfying the less-than-or-equal condition naturally satisfy the greater-than-or-equal condition, specifically on the circle's edge. Consequently, the solution of the system manifests as the full set defined by the first circle, hence revealing an infinite number of solutions spread over this area.