Problem 48 - Algebra and Trigonometry

Question

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 to the system of inequalities is the set of points that lie in the area inside and including the boundary of the circle \(x^{2}+y^{2} \leq 4\), but also above and not including the line from the inequality \(x+y > 1\).

Step 1: Identify the Form of Equations

The first inequality \(x^{2}+y^{2} \leq 4\) is a circle centered at the origin with a radius of 2. Since it is a 'less than or equal to' inequality, the solution includes the boundary of the circle and the area inside it. The second inequality \(x+y > 1\) is a linear inequality representing a line, and since it's a 'greater than' inequality, the solution will be the region above the line and not including the line itself.

Step 2: Graph the circle

Graph the circle \(x^{2}+y^{2} \leq 4\) using the center (0, 0) and radius 2. The area representing the solution will be inside the circle including the boundary.

Step 3: Graph the line

Graph the line given by \(x+y=1\). Note that since the inequality is greater than, the solution will not include the line. This can be represented by a dotted line. The solution will be the region above the line.

Step 4: Identify the solution of the system of inequalities

The solution to the system of inequalities will be the region that satisfies both inequalities. This will be the overlapping area from the two graphs. Since the second inequality does not include the boundary line, any point on the boundary line (which is included in the solution of the circle) should not be included.

Key Concepts

Inequality Solutions

When dealing with inequality solutions, we're searching for all the possible values that satisfy the conditions of the inequality. Unlike equations, inequalities do not point to a single solution but rather a range of solutions that make the inequality true.

For example, an inequality such as \(x^2 + y^2 \text{ is less than or equal to } 4\) defines a solution set within the bounds of a circle on a coordinate plane. Every point inside this circle, including its edge, makes the inequality valid. This illustrates the concept of 'inclusive' inequalities, where the equal part (\(\text{ is less than or equal to }\)) indicates that the boundary is part of the solution set.

Conversely, when an inequality does not have the equal component, such as \(x + y > 1\), it suggests that the solutions are 'exclusive' of the boundary. In this case, the region that satisfies the inequality is above the line \(x + y = 1\) but does not include the line itself. These concepts are critical in visualizing and understanding the area of overlap which represents the solution to a system of inequalities.

Graphing Inequalities

Graphing inequalities is a powerful tool to visualize the range of possible solutions. To graph an inequality, you start by drawing the equation as if it was an equality. Then, you use shading or other indicators to represent the inequality's solutions.

For a circle inequality like \(x^2 + y^2 \text{ is less than or equal to } 4\), you'll plot the circle with radius 2, centered at the origin. The inequality tells us that the region of interest is inside or on the circle; therefore, this area should be shaded to indicate the solutions. For

Dotted and Solid Lines

In the linear inequality \(x + y > 1\), you normally graph the line \(x + y = 1\) first. A solid line expresses that points on the line are included in the solutions (for 'greater than or equal to' or 'less than or equal to' inequalities). However, because our inequality specifies 'greater than', it excludes the line itself, so we draw a dotted line instead. Then, you shade the region above the line to show where the inequality is true.

By combining the graphics of both inequalities, you reveal the area where both conditions are satisfied, giving you a clear idea of the solution set.

Circle and Line Systems

When working with circle and line systems, you have a unique challenge. Circles represent a set of all points equidistant from a center, while lines divide the plane into two halves. A system comprised of a circle and a line will usually intersect in two, one, or no points at all, depending on their respective positions.

For the circle described by \(x^2 + y^2 \text{ is less than or equal to } 4\) and the line \(x + y > 1\), you find the common area that meets the conditions of both the circle and the line. In this scenario, the line may cross the circle and form a 'slice' of the circle as part of the solution area. This

Intersection of Shapes

segment shows the potential for multiple solution regions within the same system, illustrating the complexity and richness of circle and line systems in inequality graphing.

The overlapping region is the set of points that satisfy both the circle's area (inclusive of its boundary) and the area above the dotted line (exclusive of the line itself). This is why this region of overlap is crucial—it represents where both inequalities are valid concurrently.