Problem 24 - 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. $$\left\\{\begin{array}{l} x-y>0 \\ 4+y \leq 2 x \end{array}\right.$$

Step-by-Step Solution

Answer

Vertices: (2,0), (4,4), and (0,0); Solution set is bounded.

Step 1: Rewrite the Inequalities

First, rewrite the inequalities to make them easier to graph.- Inequality 1: \(x - y > 0\) becomes \(y < x\). This represents all points below the line \(y = x\).- Inequality 2: \(4 + y \leq 2x\) becomes \(y \leq 2x - 4\). This represents all points on or below the line \(y = 2x - 4\).

Step 2: Graph the Boundary Lines

Graph the lines \(y = x\) and \(y = 2x - 4\) on the coordinate plane:- The line \(y = x\) is a straight line with a slope of 1, passing through the origin. Use a dashed line for \(y < x\).- The line \(y = 2x - 4\) has a slope of 2 and y-intercept at -4. Use a solid line for \(y \leq 2x - 4\).

Step 3: Identify the Intersection Points

To find the intersection of the boundary lines, set \(y = x\) and \(y = 2x - 4\) equal to each other.Solve: \[\begin{align*}x &= 2x - 4\x &= 4\end{align*}\]Substitute \(x = 4\) back into \(y = x\) to find \(y = 4\).Thus, the intersection point is \((4, 4)\).

Step 4: Shade the Solution Region

Determine the appropriate region to shade for the inequalities: - For \(y < x\), shade below the line \(y = x\).- For \(y \leq 2x - 4\), shade below the line \(y = 2x - 4\).The solution region is where the shaded areas overlap. It should be below both lines and does not extend infinitely in any direction.

Step 5: Identify the Vertices of the Solution Region

The vertices of the solution region are the points where the boundary lines intersect and where the constraints meet the axes.- The intersection point \((4, 4)\) is a vertex.- Check for intersection with axes by setting \(y = 0\) in both equations:\(x = y = 0\) for \(y = x\) gives a point on the x-axis. - Set \(y = 0\) in \(y \leq 2x - 4\): \(0 = 2x - 4\), hence \(x=2\) and point is \((2, 0)\).

Step 6: Determine Boundedness

The solution set is a region constrained by lines in all directions without extending to infinity, thus it is bounded. The vertices are \((2,0)\), \((4,4)\), and the origin \((0,0)\).

Key Concepts

System of Inequalities

When working with a system of inequalities, you're dealing with more than one inequality at a time. In the provided exercise, there are two inequalities:

• Inequality 1: \( x - y > 0 \)
• Inequality 2: \( 4 + y \leq 2x \)

Each inequality represents a half-plane on the graph, and together they define a specific shaded region known as the solution region. The solution to a system of inequalities is where the regions of each inequality overlap. By rewriting the inequalities, we express them in a way that can be easily graphed and visualized:

• \( x - y > 0 \) becomes \( y < x \)
• \( 4 + y \leq 2x \) becomes \( y \leq 2x - 4 \)

This setup allows you to understand the critical areas to look at when solving.

Inequality Graphing

Graphing inequalities involves sketching the lines associated with each inequality and shading the region that satisfies the conditions of the inequality. This usually involves these steps:

• First, graph the lines \( y = x \) and \( y = 2x - 4 \) on the coordinate plane.
• The line \( y = x \) is represented with a dashed line since it corresponds to the inequality \( y < x \), indicating that solutions are not included on the line itself but below it.
• The line \( y = 2x - 4 \) is solid, representing the inequality \( y \leq 2x - 4 \), indicating solutions are on or below this line.

Make sure to pay attention to whether the solution set intersects with the axes, and use test points to confirm which regions to shade. Simply choose a point not on the lines and check if it satisfies the inequality to determine which side to shade.

Bounded Solution Set

A bounded solution set is crucial in understanding the nature of solutions available. It refers to a solution region that is entirely constrained and enclosed, as opposed to expanding infinitely in any direction. To determine if the solution set is bounded in this problem:

• Observe the overlapping shaded region formed by both inequalities on the graph.
• If this region doesn't extend infinitely and is encapsulated by the lines or the axes, it's bounded.
• In this exercise, the lines and axes form a triangular region, indicating that the solution is indeed bounded.

Bounded regions are significant because they provide a finite set of solutions, often representing more manageable constraints in practical applications.

Vertices of a Solution Region

Vertices are the corners or points of intersection of a solution region. They form the boundaries of the solution set. To find vertices in this system of inequalities, focus on:

• The point where the boundary lines intersect: solve the equations \( y = x \) and \( y = 2x - 4 \) simultaneously to find the vertex at \( (4,4) \).
• Additional intersections with the axes: substitute \( y = 0 \) into both equations to find additional vertices. For \( y = x \), the vertex at the origin \((0,0)\) is evident.
• When setting \( y = 0 \) in \( y \leq 2x - 4 \), you find \( x = 2 \), resulting in the vertex \( (2,0) \).

These vertices are crucial as they precisely define the boundary of the feasible region. Recognizing them helps in assessing solutions to optimization problems, such as finding maximum or minimum values subjected to constraints.