Problem 41 - Elementary Algebra Within Reach

Question

In Exercises \(41-46\), sketch the graph of the system of linear inequalities. $$ \left\\{\begin{aligned} x+y & \leq 4 \\ x & \geq 0 \\ y & \geq 0 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The feasible region for the given system of inequalities is a triangle in the first quadrant with vertices at the origin (0,0), the point where y-axis intersects the line \(x + y = 4\) (0, 4), and the point where x-axis intersects the line \(x + y = 4\) (4, 0).

Step 1: Draw the Axes

First, draw the x and y axes on a graph paper. Since all variables are non-negative, only the first quadrant is the region of interest. Draw, label and scale the axes appropriately.

Step 2: Draw Inequality 1

The first inequality is \(x + y \leq 4\). Convert it to the slope-intercept form (\(y = mx + c\)). It becomes \(y \leq -x + 4\), which is a straight line with y-intercept at 4 and slope of -1. Draw this line on the graph ensuring that it's solid (since the inequality allows for equality). Also, since \(y \leq -x + 4\), the region below the line satisfies the inequality. Shade this region.

Step 3: Draw Inequality 2

The second inequality is \(x \geq 0\). This implies that the values of \(x\) are all non-negative. So, the region to the right of the y-axis satisfies this inequality. Shade this region.

Step 4: Draw Inequality 3

The third inequality is \(y \geq 0\). This means that the values of \(y\) are all non-negative. So, the region above the x-axis satisfies this inequality. Shade this region.

Step 5: Find the Feasible Region

The feasible region is the intersection of all the shaded regions. This is the region that satisfies all the inequalities.

Key Concepts

Understanding a System of Linear Inequalities

A system of linear inequalities consists of two or more linear inequalities with the same variables. In this exercise, the system is composed of three inequalities:

• \(x + y \leq 4\)
• \(x \geq 0\)
• \(y \geq 0\)

These inequalities together define a specific region on the graph where all conditions are satisfied simultaneously. The goal is to find that common area where the solutions to all the inequalities overlap. This overlapping, shaded region is called the feasible region, and it represents all possible solutions to the system. It is essential to carefully understand each inequality's meaning and how it restricts the graph to effectively solve those systems.

Exploring the Feasible Region

In the context of linear inequalities, the feasible region is the common area on the graph where all inequalities are satisfied. Finding this region involves shading areas that comply with each individual inequality.

For the given system:

• The inequality \(x + y \leq 4\) determines that the region of interest is below or on the line \(y = -x + 4\).
• For \(x \geq 0\), the surface under consideration is to the right of the y-axis.
• Lastly, \(y \geq 0\) means the area is above the x-axis.

After graphing these, the feasible region is the overlapping shaded area on the graph, typically highlighted or enclosed distinctly. It often forms a polygon in the first quadrant, defined by the boundary lines and axes. This region implies all points within it are potential solutions, satisfying every inequality in the system.

Principles of Inequality Graphing

Graphing linear inequalities can be quite intuitive once you catch the basics. You start with graphing the associated boundary line for each inequality. Convert the inequality to an equation to find the boundary. For example, \(x + y = 4\) is the boundary for \(x + y \leq 4\).

Here's how you graph a particular inequality efficiently:

• Draw a solid line if the inequality includes equality (\( \leq \) or \( \geq \)). Use a dashed line for strict inequalities (\( < \) or \( > \)).
• Choose a test point not on the line, often (0,0), to check which side of the line to shade unless the line goes through the origin.
• If the test point satisfies the inequality, shade the side of the line where the point lies. If not, shade the opposite side.

Through these steps, each inequality can visualized on the coordinate plane, guiding you to find the intersection, or the feasible region.

Using Quadrant Analysis

Quadrant analysis plays a vital role in graphing inequalities, especially in systems with variables constrained to positive values. The Cartesian plane is divided into four quadrants:

• First Quadrant: both x and y values are positive.
• Second Quadrant: x values are negative, y values are positive.
• Third Quadrant: both x and y values are negative.
• Fourth Quadrant: x values are positive, y values are negative.

Since the inequalities \(x \geq 0\) and \(y \geq 0\) restrict our solutions to the first quadrant, this limits the graphing challenge to non-negative solutions. Only the first quadrant is relevant to study in this specific system of inequalities.

This focused analysis simplifies the problem by excluding other portions of the graph, making it easier to visualize and solve the system accurately. By only considering the practical quadrant, students can better represent the feasible region and learn to target specific areas in inequality graphing.