Problem 52 - College Algebra

Question

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} (x+1)^{2}+(y-1)^{2}<16 \\ (x+1)^{2}+(y-1)^{2} \geq 4 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of inequalities is the region between the two circles with center (-1,1) and radii 2 and 4.

Step 1: Graph the first inequality

Plotting the first inequality, \((x+1)^{2}+(y-1)^{2}<16\), gives a circle with center (-1,1) and radius equal to 4. Since the circle inequality is less than 16, we shade inside the circle.

Step 2: Graph the second inequality

Plot the second inequality, \((x+1)^{2}+(y-1)^{2} \geq 4\). This will also give a circle with the same center (-1,1), but radius equal to 2. This inequality indicates that the area includes the boundary of the circle as well as outside of it.

Step 3: Find the intersection of two inequalities

The solution to the system of inequalities is the area that satisfies both the above inequalities. This results in the region between these two circles.

Key Concepts

Inequality Graphing

Graphing an inequality involves plotting the region of a coordinate plane that satisfies the inequality's condition. Unlike equations, which typically have a line or curve as their representation, inequalities are represented by shaded areas. In the context of a system involving circular inequalities, such as \( (x+1)^{2}+(y-1)^{2}<16 \), visualization is key in understanding the solution set.

When graphing the inequality, remember to start with the equality part, which typically forms the boundary. In our example, the equality would be the circle itself, which is the boundary of the inequality. Next, determine whether to shade inside or outside the boundary. For inequalities with '<' or '<=', shade inside the curve. For '>', or '>=', shade outside. Added technicality comes with the 'strict' inequalities (without the equals part), where the boundary is a dashed line, meanwhile for 'non-strict' inequalities (with the equals part) it is a solid line, indicating that points on the line are also part of the solution set. Students often find it beneficial to practice plotting points around the circle to determine the correct area to shade.

System of Inequalities

When dealing with a system of inequalities, we're essentially looking for a common area that satisfies all inequalities in the system. To graph a system, each inequality is graphed separately, and then the overlapping region that satisfies all conditions is identified as the solution.

Here are some tips to ensure clarity when graphing a system:

• Use different shading patterns or colors for each inequality to clearly distinguish them.
• Always start with a clear sketch of each individual inequality.
  
• Look for the intersection area where all conditions are met, which represents a common solution to all inequalities.
• Check a point in the shaded region to ensure it satisfies all the inequalities in the system.

A system like \( (x+1)^{2}+(y-1)^{2} \geq 4 \) can be thought of as setting restrictions, or 'fencing in', where possible solutions can lie. Drawing such a system properly promotes understanding of the composite solution set.

Circle Inequalities

Circle inequalities are specific forms of inequalities where the solution set forms a circular area or a ring on a coordinate plane. The general form of a circle equation is \( (x-h)^{2}+(y-k)^{2}=r^{2} \), where \( (h, k) \) are the coordinates of the center, and \( r \) is the radius. When this equation becomes an inequality, such as \( (x+1)^{2}+(y-1)^{2}<16 \) or \( (x+1)^{2}+(y-1)^{2} \geq 4 \), the entire interior or exterior of the circle becomes relevant, depending on the inequality sign.

To efficiently graph circle inequalities, ensure you can:

• Determine the center and radius from the inequality.
• Decide if you will need to shade the inside or outside of the circle based on the direction of the inequality sign.
• Remember that 'less than' translates to shading inside the circle, while 'greater than' means shading the area outside.

When a system has two circle inequalities with the same center, as seen in the given exercise, their combined solution set often appears as a ring or a 'donut-shaped' region, demonstrating the area sandwiched between the two boundaries.