Problem 75 - College Algebra Essentials

Question

Without graphing, 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 system of inequalities has infinitely many solutions.

Step 1: Interpret the Inequalities

Given are two inequalities: \((x+4)^{2}+(y-3)^{2} \leq 9\) and \((x+4)^{2}+(y-3)^{2} \geq 9\). The inequalities \((x+4)^{2}+(y-3)^{2} \leq 9\) and \((x+4)^{2}+(y-3)^{2} >= 9\) represent a filled circle with center at (-4,3) and having radius 3, and an exterior region outside the boundary of a circle with same center and radius respectively.

Step 2: Determine Possible Solutions

Interpreting the inequalities geometrically, we can determine the feasible region of solutions. Points that lie within or on the boundary of the circle are solutions of the first inequality. Points that lie outside or on the boundary of the circle are solutions of the second inequality. Thus, the system will have solutions common to both inequalities.

Step 3: Conclusion

As both the inequalities share the boundary of the circle as common solution, the system of inequalities \((x+4)^{2}+(y-3)^{2} \leq 9\) and \((x+4)^{2}+(y-3)^{2} \geq 9\) have infinitely many solutions, represented by the points on the boundary of the circle with center (-4, 3) and radius 3.

Key Concepts

Geometric Interpretation

Understanding the geometric interpretation is key to grasping systems of inequalities. When you see inequalities like \((x+4)^{2}+(y-3)^{2} \leq 9\) and \((x+4)^{2}+(y-3)^{2} \geq 9\), they indicate regions on the coordinate plane. These inequalities involve circles centered at (-4, 3) with radius 3.
The inequality \((x+4)^{2}+(y-3)^{2} \leq 9\) describes a filled circle. This means all points within and on the circle's boundary satisfy the inequality. On the other hand, \((x+4)^{2}+(y-3)^{2} \geq 9\) describes the area outside and including the circle's boundary.
Visualizing this, imagine a circle drawn around point (-4, 3) with a radius reaching out to 3 units. Anything inside or touching this circle fulfills the first condition (\(\leq 9\)). Meanwhile, anything touching or outside meets the second condition (\(\geq 9\)).
Together, these explain how inequalities can represent geometric forms on a graph and guide us to their solutions.

Feasible Region

The feasible region plays a significant role in solving systems of inequalities. It refers to the set of all points that satisfy every inequality in the system.
In this case, for the inequalities \((x+4)^{2}+(y-3)^{2} \leq 9\) and \((x+4)^{2}+(y-3)^{2} \geq 9\), the feasible region is where both conditions are met.
To find it, consider:

• The first inequality, \((x+4)^{2}+(y-3)^{2} \leq 9\), includes all points inside and on the circle.
• The second inequality, \((x+4)^{2}+(y-3)^{2} \geq 9\), covers everything on and outside the circle.

Thus, the feasible region where both conditions overlap is exactly the boundary of the circle. Each point on this circle at (-4, 3) with a radius of 3 is part of the feasible region, providing a rich set of solutions.

Infinitely Many Solutions

In specific systems of inequalities, like the one we're dealing with, the concept of infinitely many solutions becomes crucial. When inequalities suggest a shared region where points meet all conditions, we discover that the solution set isn't just a few isolated points, but can cover an entire area.
For the system \((x+4)^{2}+(y-3)^{2} \leq 9\) and \((x+4)^{2}+(y-3)^{2} \geq 9\), the boundary of the circle is common to both inequalities.
This indicates that every single point on this circular boundary fits perfectly inside the solution set. Since a circle consists of countless points along its edge, this means there are infinitely many solutions.
Remember:

• A system with a shared geometric boundary often leads to infinite solutions.
• All points on such a boundary satisfy both conditions together, expanding the number of solutions infinitely.

This realization allows us to appreciate the vastness of possible solutions and reinforces the power of geometric insights in systems of inequalities.