Problem 47 - College Algebra

Question

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

Step-by-Step Solution

Answer

The solution to the system of inequalities is the area inside the circle with radius 4 units and above the line \(x+y=2\).

Step 1: Graph the First Inequality

Draw a circle centered at the origin (0,0) with a radius of 4 units. The inequality \(x^{2} + y^{2} \leq 16\) means all the points inside and on the circle are included in the solution.

Step 2: Graph the Second Inequality

Draw a straight line \(x + y = 2\), using the the y-intercept method you could use two points (2, 0) and (0, 2). The line divides the coordinate plane into two regions. As the inequality is \(x + y > 2\), we take a test point on any one side of the line. If the inequality holds, then that region is our solution. If not, the other region is our solution. For example, suppose we take (0,0), it doesn't satisfy the inequality. Therefore, the region that doesn't include (0,0) is the solution, which is above the line.

Step 3: Identify the Overlapping Region

The common solution to the system of inequalities is the region where the solutions to both inequalities overlap. In this case, it's the area inside the circle and above the line that represents the solution to the system of inequalities.

Key Concepts

System of Inequalities

Understanding a system of inequalities involves looking at multiple inequalities at once and finding the common set of solutions that satisfy all the inequalities involved. Unlike a system of equations that looks for exact points of intersection, the system of inequalities seeks a region where the conditions of all inequalities are met.

In the given exercise, you have two inequalities: a circle representing all the points up to a certain distance from the center, and a linear inequality that separates the plane into two regions. To solve this system, you need to graph both on the same plane and look for the area that satisfies both at the same time.

Here are a few pointers to improve your understanding:

• Identify what each inequality represents in terms of the area on a graph.
• Overlapping regions are key to the solution.
• Remember, the solution is often a shared area, not just a line or a single point.

When you master these concepts, solving systems of inequalities becomes a task of finding where two or more areas come together on the graph.

Inequality Solution

An inequality solution on the coordinate plane is not just a point, but an entire area that meets the condition set by the inequality. It is essential to know how to identify these solution regions.

For the circle inequality \(x^2 + y^2 \leq 16\), any point whose distance from the origin is less than or equal to 4 will be a solution. To determine the solution region of \(x + y > 2\), you can use a 'test point' that is not on the line \(x + y = 2\) to see which side of the line is part of the solution.

Consider these tips:

• Always test a point to confirm the correct side of a line inequality.
• The equal part of '\(\leq\)' and '\(\geq\)' means the line itself is included in the solution.
• Use dashed or solid lines to depict whether the boundary is included (solid) or not (dashed).

By understanding how to find and represent these solutions, you will be able to graphically solve any inequality.

Coordinate Plane Graphing

Graphing on the coordinate plane is not only about plotting points, but also involves drawing lines, curves, and shading regions to represent solutions to inequalities. Accuracy and neatness are crucial for correct interpretation.

To graph the inequality \(x^{2} + y^{2} \leq 16\), draw a circle with radius 4. For \(x + y > 2\), draw a line and shade the region above it to represent all points that make the inequality true.

Here's how to excel in coordinate plane graphing:

• Use a ruler to draw straight lines accurately.
• For circles or curves, use a compass or a round object to trace.
• Shade lightly so overlapping regions are clearly visible.
• Label your graph to help interpret the regions correctly.

Practice graphing with attention to detail to ensure your solutions are clear and precise. This is essential when dealing with multiple inequalities.