Problem 22 - Algebra and Trigonometry

Question

19–40 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 are (4, 4), (0, 0), and (0, -4). The solution set is bounded.

Step 1: Rewrite Inequalities

First, rewrite the inequalities in slope-intercept form. For the inequality \(x - y > 0\), rewrite as \(y < x\). For the inequality \(4 + y \leq 2x\), rewrite as \(y \leq 2x - 4\).

Step 2: Graph Both Lines

Graph the line \(y = x\) as a dashed line because the inequality is \(<\), indicating that points on the line are not included. Graph the line \(y = 2x - 4\) as a solid line because the inequality is \(\leq\), meaning points on the line are included.

Step 3: Determine Shaded Regions

For the inequality \(y < x\), shade below the line because any point below the line satisfies \(y < x\). For the inequality \(y \leq 2x - 4\), shade below or on the line because any point below or on the line satisfies \(y \leq 2x - 4\).

Step 4: Identify Intersection of Shaded Regions

Identify the region where both shadings overlap. This overlapping region is the solution set for the system of inequalities. The lines intersect at a point, which is one of the vertices of the solution set.

Step 5: Find Intersection Point

To find the coordinates of the intersection, set the equations \(y = x\) and \(y = 2x - 4\) equal: \(x = 2x - 4\). Solve to get \(x = 4\), and substitute back into \(y = x\) to find \(y = 4\). So, the intersection point is \((4, 4)\).

Step 6: Identify Vertices of the Solution Set

The vertices are at the points \((x, y)\) where the solution space is bounded by either line and the coordinate axes. The y-intercept of the line \(y = 2x - 4\) is \(-4\), so another vertex is \((0, -4)\). The x-axis intercept of \(x = y\) is at the origin \((0, 0)\). Thus, vertices are \((4, 4)\), \((0, 0)\), and \((0, -4)\).

Step 7: Determine if the Solution Set is Bounded

Since the solution region is defined by a polygon with finite sides connecting points \((4, 4)\), \((0, 0)\), and \((0, -4)\), the solution set is bounded.

Key Concepts

Graphing Inequalities

When dealing with a system of inequalities, we graph each inequality on the same coordinate plane to understand where the solutions to these inequalities overlap. This process helps us visualize all the solutions visually and determine the area these solutions occupy.
Firstly, for inequalities like the ones given in the problem, they need to be rewritten in a way that makes them easier to graph, typically slope-intercept form (\(y = mx + b\)).

• The inequality \(x - y > 0\) gets converted to \(y < x\), making a line we will graph as \(y = x\).
• Similarly, \(4 + y \leq 2x\) is rewritten as \(y \leq 2x - 4\).

Drawing these lines correctly is crucial. For \(y = x\), a dashed line represents an inequality \(<\), meaning points exactly on the line are not included in solutions. For \(y \leq 2x - 4\), a solid line is used, showing points on the line are part of the solution. After graphing these, we shade the regions that satisfy each inequality, carefully noting which side of each line meets the criteria.

Bounded Solution Set

A solution set is considered bounded when it is enclosed within a finite area on the graph. In this problem, we graphically determine whether the solution set is bounded by examining the overlap of shaded regions from each inequality.
The inequalities create a shape or polygon in which all interior points satisfy both conditions of the system. This region is called the solution set.

• If this region is completely enclosed by the graph lines, not extending infinitely in any direction, it is considered bounded.

In this exercise, the region where the shaded areas overlap forms a triangle between the points \((4, 4)\), \((0, 0)\), and \((0, -4)\). Because this region does not extend beyond these vertices, it is a bounded solution set. It helps students to visualize bounded regions as shapes distinctly formed by the intersection of graph lines, often with clear boundaries from vertices.

Intersection of Lines

Finding the intersection of lines is essential in determining vertices of the solution set for a system of inequalities. Intersections are the points where lines cross each other, representing solutions that satisfy both equations in the system.
To find points of intersection, set the equations of the two lines equal to each other and solve for \(x\).

• By solving \(y = x\) and \(y = 2x - 4\), we equate them: \(x = 2x - 4\).
• This simplifies to \(x = 4\).
• Substitute \(x = 4\) back into either equation to find \(y = 4\).

The intersection point is \((4, 4)\), which is one of the solution set's vertices.
Knowing the intersection points helps us draw and validate the vertices of the bounded area, thus visualizing where our solution set lies, as it is the area formed between intersection points. This understanding is key in solving systems of inequalities graphically.