Problem 48 - College Algebra

Question

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

Step-by-Step Solution

Answer

The solution set is the region where both solution areas for the inequalities overlap. It is the area within the circle of radius 2 and above the line \(x+y>1\).

Step 1: Graphing the inequality \(x^{2}+y^{2} \leq 4 \)

This equation is in the form of a circle with center at the origin and radius of 2, since \(\sqrt{4}=2\). Plot a circle with radius 2 at the origin (0,0), and it includes the area inside of the circle because of the \(\leq\) sign in the inequality.

Step 2: Graphing the inequality \(x+y>1\)

This equation represents a straight line, which is the equation in the form of \(y = -x+1\). Plot this line on the same graph. The solution for this inequality will be the area above this line as indicated by the > sign in the inequality.

Step 3: Identifying the solution area

The solution for the system is the area where both these solution areas overlap. This is the area inside the circle and above the line.

Key Concepts

systems of inequalities

Systems of inequalities involve multiple inequalities that you need to solve or graph at the same time. Each inequality in the system describes a condition that a point or set of points must satisfy. When dealing with systems of inequalities, you're looking to find a region on the graph where all inequalities hold true.

To solve a system of inequalities graphically, each inequality is represented by a distinct area on a graph. The solution to the system is where these areas intersect. This can encompass one or more regions or, in some cases, reveal that no such region exists if the system has no solution.

• First, understand each inequality individually.
• Graph each inequality on the same coordinate plane.
• Identify the regions that satisfy all inequalities simultaneously.

By treating each inequality as a condition, graphing systems of inequalities shows you vividly how different conditions can coexist or conflict.

coordinate plane

The coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves by using two numbers (coordinates). These coordinates denote the horizontal (x-axis) and vertical (y-axis) positions of points. When graphing inequalities, the coordinate plane becomes your canvas to visualize solutions.

In our specific exercise, the coordinate plane helps us:

• Plot a circle centered at the origin representing the first inequality \(x^2 + y^2 \leq 4\).
• Draw a line representing the second inequality \(x + y > 1\).

As you work through plotting, always remember:

• The x-axis runs horizontally, and the y-axis runs vertically.
• Use these axes to determine and locate any point with an (x, y) coordinate.

Understanding the layout of the coordinate plane is crucial in analyzing and interpreting systems like these.

solution set

A solution set in the context of systems of inequalities refers to the collection of all points that satisfy every equation or inequality within the system. Finding a solution set involves both graphing and analyzing where different shaded areas overlap in the coordinate plane.

For the given system:

• The inequality \(x^2 + y^2 \leq 4\) includes all points inside and on the circle of radius 2 centered at the origin.
• The inequality \(x + y > 1\) includes all points above the line \(y = -x + 1\).
• The solution set is the overlap between these two sets; it's the region inside the circle and above the line.

The solution set can sometimes be a complex shape, often confined to specific regions on the graph. For ease, you can emphasize these regions by shading or coloring on the coordinate plane.

overlapping regions

When dealing with systems of inequalities, the goal is often to identify overlapping regions. This is the area where the solutions to the individual inequalities coincide. To successfully pinpoint these overlaps, follow these steps:

1. Graph each inequality one by one to visualize their respective regions.
2. Carefully observe the graph for areas where these regions intersect or overlap.
3. Highlight these intersections; they represent your solution set.

In the example system, the overlapping region is inside the circle (from the inequality \(x^2 + y^2 \leq 4\)) and above the line (stemming from \(x + y > 1\)).

Understanding overlapping regions helps you determine where all conditions of the system are satisfied and is key to solving these exercises correctly. Consider them as the focal point of your graph, embodying the solution to the system.