Problem 51 - College Algebra Essentials

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} (x-1)^{2}+(y+1)^{2}<25 \\ (x-1)^{2}+(y+1)^{2} \geq 16 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution set of the given system of inequalities is the annulus or ring shaped region bounded by the two circles with radii 4 and 5, centered at (1, -1).

Step 1: Understanding the inequalities

The given inequalities are formula of circles: \( (x-h)^2 + (y-k)^2 = r^2 \), where (h, k) is the center and r is the radius but we have inequality signs rather than '=' which means these inequalities are for areas that circles cover. For the first equation, the circle is centered at (1, -1) and has a radius of 5 (since \( \sqrt{25} = 5 \) ). For the second equation, the circle is also centered at (1, -1) but has a smaller radius 4 ( \( \sqrt{16} = 4 \) ). The '<' sign indicates inside the circle, while '≥' indicates the border point of the circle and the region outside it.

Step 2: Graphing the inequalities

First draw the circles on a graph: the larger one with a radius of 5 units and smaller one with a radius of 4 units both centered at (1, -1). For the larger circle use dashed line because it is just less than, not equals to, but for the smaller circle use a solid line because it is greater than or equals to. Now, shade the area inside the larger circle and outside the smaller circle, this shaded region should look like a doughnut or ring shape which represents the area that satisfies both inequalities.

Step 3: Find the solution set

The area that our solution set lies will be the overlapping region of the two areas we defined above. Observing the graph, it can be seen that these two regions overlap in a ring shaped area (an annulus) bounded by the two circles. This is the common solution area for both inequalities and hence, it is the solution set for the given system of inequalities.

Key Concepts

Solution Set

The solution set of a system of inequalities is the collection of all points that satisfy all the inequalities in the system. It's like a Venn diagram where you find the common region that meets all conditions in your math problem.

In our example, we are dealing with two inequalities representing circles. The solution set here is the area where the conditions of both inequalities overlap. To determine this, you graph each inequality. First, graph the larger circle with the radius of 5 units using a dashed line, which represents "less than" (i.e., points inside the circle). Next, for the smaller circle with a radius of 4 units, use a solid line, representing "greater than or equal to" (i.e., points on or outside the circle).

Shade the intersection area carefully between these two graphical representations. The shared or overlapping zone appears like a ring (annular) region between the two circles. This ring forms the solution set for the system. It's essential because it visually shows all the potential solutions that fulfill both inequalities simultaneously.

System of Inequalities

A system of inequalities involves more than one inequality that you consider simultaneously. Solving such systems often requires not just finding a solution but identifying the set of all possible solutions, hence a solution set.

To solve, you need to:

• Graph each inequality on the coordinate plane. This involves plotting the boundary as a solid or dashed line based on the inequality type (e.g., '>=' or '<').


• Identify the region that meets the criteria of all inequalities. This might be challenging but can often visually observed by the overlap of the shaded regions.

Circles in our example function within this system. The first inequality defines a region inside a circle with the dashed border (r = 5), and the second defines the exterior along with the border of a smaller circle (r = 4). By graphically overlapping these areas, the system of inequalities allow you to see which region meets both conditions, given as an annulus shape.

Circles in Coordinate Geometry

Circles in coordinate geometry can be defined by the equation \[(x - h)^2 + (y - k)^2 = r^2\]where

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

In inequalities,

• When \[(x - h)^2 + (y - k)^2 < r^2\], it represents the area inside the circle.


• When \[(x - h)^2 + (y - k)^2 \geq r^2\], it indeed includes the boundary and everything outside the circle.

In our example, two circles share the same center (1, -1) but different radii. The inequality signs define our area of interest. We graph one with a radius of 5 units and the other of 4 units. Visualizing these on the coordinate geometry allows us to perceive how inequalities operate within circular regions, thereby leading us to identify the solution set or region where the conditions of all inequalities are satisfied.