Problem 67 - A Graphical Approach to Precalculus with Limits

Question

Graph the solution set of each system of inequalities by hand. $$\begin{array}{r}x \leq 4 \\\x \geq 0 \\\y \geq 0 \\\x+2 y \geq 2\end{array}$$

Step-by-Step Solution

Answer

Shade the region bounded by \(0 \leq x \leq 4\), \(y \geq 0\), and \(x + 2y \geq 2\).

Step 1: Understand the Inequalities

To graph the solution set, we need to understand each inequality: \(x \leq 4\) represents all points to the left of or on the vertical line \(x = 4\); \(x \geq 0\) represents all points to the right of or on the vertical line \(x = 0\); \(y \geq 0\) includes all points above or on the horizontal line \(y = 0\); \(x + 2y \geq 2\) is a line with the equation \(y \geq -\frac{1}{2}x + 1\). Points above or on this line satisfy the inequality.

Step 2: Graph the Line x = 4

Draw a vertical line at \(x = 4\) on the coordinate plane. This line acts as a boundary for the inequality \(x \leq 4\), shading to the left of the line as this is the region of interest.

Step 3: Graph the Line x = 0

Draw a vertical line at \(x = 0\) on the coordinate plane. Shade to the right of the line, as this satisfies \(x \geq 0\). This line corresponds to the y-axis.

Step 4: Graph the Line y = 0

Draw a horizontal line along the x-axis, which corresponds to \(y = 0\). We shade above this line because \(y \geq 0\).

Step 5: Graph the Line x + 2y = 2

To draw this line, find two points on the line by setting \(x = 0\) and solving for \(y\) which gives \(y = 1\), and \(y = 0\) to find \(x = 2\). Plot both points (0,1) and (2,0), and connect them to form the line. Shade above this line as this region satisfies \(x + 2y \geq 2\).

Step 6: Identify the Feasible Region

The feasible region is where all the shaded areas overlap. Check that this includes the intersection of all the boundaries set by the inequalities: within \(0 \leq x \leq 4\), \(y \geq 0\), and above or on the line \(y \geq -\frac{1}{2}x + 1\).

Step 7: Shade the Solution Set

Shade the region where \(0 \leq x \leq 4\), \(y \geq 0\), and \(x + 2y \geq 2\) overlap. This is the area that satisfies all given inequalities.

Key Concepts

Graphical Solutions

In order to solve a system of inequalities graphically, you must translate each inequality into a visual boundary on a coordinate plane. Each inequality defines a specific region. The goal is to illustrate these regions on the graph and identify their intersections.

• Start by analyzing each inequality separately. Notice whether it describes a vertical or horizontal line, or if it forms an oblique line.
• Plot each line using standard graphing techniques like finding intercepts or using slope-intercept form.
• Remember that each inequality represents a half-plane, meaning that one side of the plotted line will satisfy the inequality.

Drawing each line and shading the appropriate region helps visualize where the solutions to the inequalities overlap. This overlapping region represents the set of solutions that satisfy all conditions given in the system, forming the basis of the graphical solution method.

Feasible Region

The feasible region is a crucial concept when graphing systems of inequalities. It represents the area on a graph that satisfies all inequalities at the same time. To find the feasible region:

• Graph each inequality following the steps for graphical solutions, including plotting lines and shading the respective regions.
• Observe the areas of overlap. This intersection is the feasible region, where all conditions of the system of inequalities coincide.
• This region may be bounded if all inequalities restrict it to a finite area, or unbounded if some inequalities extend indefinitely.

Understanding the feasible region is vital in contexts like optimization problems where you need to find the maximum or minimum values within this region. Not every point in the plane will satisfy all inequalities, only those within this shared, feasible area.

Shading Techniques

Shading techniques are fundamental in visualizing systems of inequalities. Shading specifies which side of a boundary line is part of the solution set for an inequality. Here's how you can effectively use shading techniques:

• After graphing a line, decide whether to shade above, below, to the left, or right based on the inequality's greater than or less than sign.
• For a condition like \(x \leq 4\), shade everything leftward from the line \(x = 4\). For \(y \geq 0\), shade above the \(x\)-axis.
• Use consistent shading patterns to avoid confusion, especially when multiple boundaries intersect. Cross-hatch or use different colors for clarity.
• Where all shadings meet is the feasible region, important for identifying valid solutions.

Effective shading illustrates regions that work for each inequality, enabling a clear understanding of how different constraints interact. It highlights the viable areas swiftly, making problem-solving more intuitive.