Problem 22 - Precalculus: Mathematical 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: (4, 4); Solution set is unbounded.

Step 1: Convert Inequalities to Equations

To graph the inequalities, first convert them to equations. This helps in identifying boundary lines.1. From the inequality \(x - y > 0\), the equation is \(x - y = 0\).2. From the inequality \(4 + y \leq 2x\), the equation is \(4 + y = 2x\).

Step 2: Graph the First Linear Equation

Graph the equation \(x - y = 0\). This line passes through the origin with a slope of 1.- Intercepts: x-intercept at (0,0) and y-intercept at (0,0).- The line divides the plane into two regions: above the line (where \(x - y > 0\)) and below the line (where \(x - y < 0\)).- Since the inequality is \(x - y > 0\), shade the region above the line.

Step 3: Graph the Second Linear Equation

Graph the equation \(4 + y = 2x\). This line also passes through the point where x and y values satisfy the equation.- Rearrange to \(y = 2x - 4\) to identify slope-intercept form (slope = 2, y-intercept = -4).- Find two points: when x=0, y=-4 (0, -4); and when y=0, x=2 (2, 0).- The line divides the plane into two regions: above the line (where \(4 + y < 2x\)) and below the line (where \(4 + y > 2x\)).- Since the inequality is \(4 + y \leq 2x\), shade the region below the line.

Step 4: Determine the Solution Region

The solution region is the intersection of the shaded regions from both inequalities.1. The solution to \(x - y > 0\) is above the line \(x = y\).2. The solution to \(4 + y \leq 2x\) is below the line \(y = 2x - 4\).3. Find regions of overlap: shade where both conditions are true.

Step 5: Find the Vertices of the Solution Region

Identify points of intersection where lines intersect each other or the axes.1. Intersection of \(x - y = 0\) and \(y = 2x - 4\): - Substitute \(y = x\) into \(y = 2x - 4\): \[ x = 2x - 4 \] \[ 4 = x \] Substituting back, \(y = 4\). So the vertex is (4, 4).2. The solution does not intercept the axes further since they don't form closed figure on axes.

Step 6: Determine if the Solution Set is Bounded

Analyze the solution region to determine if it is bounded or unbounded. - Only one vertex intersection is found, and the region extends indefinitely without bounds in one direction. - Conclusion: The solution set is unbounded.

Key Concepts

Graphing Inequalities

Graphing inequalities involves transforming them into graphs that show all points that satisfy the inequalities. This includes identifying boundary lines using corresponding equations. Start by converting each inequality into a linear equation. For example, the inequality \(x - y > 0\) becomes \(x - y = 0\).

Draw this equation on the graph as a line. Determine which side of this line satisfies the inequality. Since the inequality is strict \(x - y > 0\), you will shade the area above the line where all \((x, y)\) pairs meet the condition. Similarly, turn \(4 + y \leq 2x\) into \(4 + y = 2x\) and graph it. This equation, rearranged to \(y = 2x - 4\), shows a line with a slope of 2 and y-intercept at (-4). For this inequality, shade beneath the line since the inequality is \(\leq\).

The intersection of the shaded regions represents all solutions common to both inequalities.

Linear Equations

Linear equations are the foundations for graphing inequalities. They express a straight line in the coordinate plane. Equations like \(x - y = 0\) and \(y = 2x - 4\) form these lines.

To effectively graph a linear equation, first convert it to the slope-intercept form \(y = mx + b\). Then, identify the slope \(m\) and the y-intercept \(b\). For \(x - y = 0\), rewrite it as \(y = x\), indicating a slope of 1 and passing through the origin (0,0).

Compare this to \(y = 2x - 4\) where the slope \(m\) is 2, moving at a steeper angle intercepting the y-axis at (0, -4). These equations divide the plane, giving different solution regions for corresponding inequalities.

Bounded and Unbounded Solution Sets

Understanding bounded and unbounded solution sets helps in realizing the nature of the feasible region for a system of inequalities. If the solution region is entirely enclosed within a boundary, it is considered bounded. However, if the feasible area stretches indefinitely without forming a closed boundary, it is unbounded.

In our example, when we determine intersections such as where \(x - y = 0\) meets \(y = 2x - 4\), we find a single vertex at (4, 4). This, along with the shading, signifies that the solutions extend infinitely in one direction without forming a closed shape.

Thus, this system has an unbounded solution set—a common scenario in linear programming where not all constraints can close off the solution region.