Problem 33 - Precalculus Mathematics for Calculus

Question

Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{aligned} x^{2}+y^{2} & \leq 4 \\ x-y &>0 \end{aligned}\right.$$

Step-by-Step Solution

Answer

Vertices: \((\sqrt{2}, \sqrt{2}), (-\sqrt{2}, -\sqrt{2})\); Solution set is bounded.

Step 1: Understand the inequalities

We have two inequalities: (1) the inequality \(x^2 + y^2 \leq 4\) which represents a circle centered at the origin with radius 2, and (2) \(x - y > 0\) which represents the region above the line \(y = x\).

Step 2: Graph the circle

To graph the inequality \(x^2 + y^2 \leq 4\), draw the circle with a radius of 2 centered at the origin \((0,0)\). The solution includes all the points on and inside this circle.

Step 3: Graph the line

The inequality \(x - y > 0\) is equivalent to \(y < x\). Graph the line \(y = x\) on the same coordinate plane. The solution will be the area below this line.

Step 4: Find the region of intersection

The solution to the system of inequalities is the region where the two graphs overlap. Shade the region inside the circle and below the line \(y = x\).

Step 5: Determine the vertices

Find where the line \(y = x\) intersects the circle \(x^2 + y^2 = 4\). Solve the system \(x = y\) and \(x^2 + y^2 = 4\) simultaneously: substitute \(x = y\) into the circle's equation to get \(2x^2 = 4\), yielding \(x = \pm \sqrt{2}\). Thus, the vertices are \((\sqrt{2}, \sqrt{2})\) and \((-\sqrt{2}, -\sqrt{2})\).

Step 6: Check if the solution is bounded

The solution set is bounded because it is the interior of the circle constrained by the line \(y = x\), resulting in a finite area.

Key Concepts

Graphing Inequalities

Graphing inequalities involves showcasing the regions where inequalities hold true on a coordinate plane. When you have an inequality, such as \(x^2 + y^2 \leq 4\), it means you're looking for all points \((x, y)\) that satisfy the inequality.

For the inequality \(x^2 + y^2 \leq 4\),

• This represents a circle centered at the origin \((0,0)\).
• The radius of this circle is 2 since the equation of the circle without the inequality symbol is \(x^2 + y^2 = 4\).
• The inequality symbol \(\leq\) indicates that points on the circle, as well as those inside the circle, are included in the solution.

For the inequality \(x - y > 0\), it translates to \(y < x\), which involves:

• A line that goes through the origin, with a slope of 1, represented by the equation \(y = x\).
• The area of interest is below the line, as the inequality \(y < x\) indicates points below the line are solutions.

By graphing inequalities, you illustrate these regions to find their intersection.

Bounded Solution Set

A bounded solution set refers to a solution-holding area that is limited or enclosed in space. Think of it like a small fenced area, where the solutions can't escape the borders.

In this context, the solution to the system of inequalities involves finding where these bounded areas overlap.

The circle \(x^2 + y^2 \leq 4\), bounds the solution because:

• It has a finite radius, limiting the solution to its interior.
• The solution region can't extend past this circle's radius of 2, making it bounded.

The line \(y = x\) further confines the solution by cutting off part of this circle, specifically keeping the valid solutions below this line.

• This combination ensures the solution set is not only enclosed by the circle but also has a finite and distinct area on one side of the line.

A bounded solution set means that the overlapping region is finite and well-defined within the circle's closure.

Intersection of Graphs

The intersection of graphs is the overlapping region where solutions to a system of inequalities exist. To find this, you need to determine where the regions defined by each inequality meet on the graph.

For the system given:

• The circle \(x^2 + y^2 \leq 4\) includes all points on and inside its boundary.
• The line \(y = x\) divides the plane into two regions, where \(y < x\) is the area that satisfies the second inequality.

To find the intersection:

• Identify the shading where the circle and the area below the line overlap.
• The vertices, where the line intersects the circle, are important as they often define the corners of the solution region.

The points where the graphs intersect give the boundaries of the solution set, and these precise locations or points (vertices) help to confirm that the solution is entirely contained within the intersection area.