Problem 51 - A Graphical Approach to College Algebra

Question

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} 3 x-2 y & \geq 6 \\ x+y & \leq-5 \\ y & \leq 4 \end{aligned}$$

Step-by-Step Solution

Answer

Shade the overlapping region where all inequalities are satisfied on the graph.

Step 1: Rewrite inequalities for graphing

We start by rewriting each inequality in a form that is easier to graph, specifically the slope-intercept form of a linear equation (i.e., \(y = mx + b\)). This will make it easier to identify the boundary lines and shaded regions:1. For \(3x - 2y \geq 6\), rewrite in slope-intercept form as \(y \leq \frac{3}{2}x - 3\).2. For \(x + y \leq -5\), rewrite in slope-intercept form as \(y \leq -x - 5\).3. The third inequality, \(y \leq 4\), is already in slope-intercept form.

Step 2: Graph boundary lines

Graph each of the linear equations obtained from the inequalities on the coordinate plane by treating the inequalities as equalities to get the boundary lines:1. Graph the line \(y = \frac{3}{2}x - 3\), which has a slope of \(\frac{3}{2}\) and y-intercept of -3.2. Graph the line \(y = -x - 5\), which has a slope of \(-1\) and y-intercept of -5.3. Graph the horizontal line \(y = 4\).

Step 3: Determine line styles

Determine if the lines should be solid or dashed:1. For \(3x - 2y \geq 6\) (\(y \leq \frac{3}{2}x - 3\)), use a solid line because the inequality includes equality (\(\geq\)).2. For \(x + y \leq -5\) (\(y \leq -x - 5\)), use a solid line because the inequality includes equality (\(\leq\)).3. For \(y \leq 4\), use a solid line because the inequality includes equality (\(\leq\)).

Step 4: Shade the feasible regions

Shade the regions that satisfy each inequality:1. For \(y \leq \frac{3}{2}x - 3\), shade below the line.2. For \(y \leq -x - 5\), shade below the line.3. For \(y \leq 4\), shade below the line.The solution to the system of inequalities is the region where all shaded areas overlap.

Step 5: Identify the solution set

The solution to the system is the region where all shaded areas intersect. It represents the set of points that satisfies all three inequalities. Graphically determine this area and identify it clearly on the graph. Ensure the overlapping region is visibly distinct from other shaded areas.

Key Concepts

Graphing Inequalities

Inequalities express a range of values rather than an exact value and are crucial in various mathematical and real-world applications. When graphing a system of inequalities, you are looking for the set of values that simultaneously satisfy all inequalities in the system. Each inequality can be transformed and illustrated on a coordinate plane as a half-plane. This half-plane represents all solutions to the inequality.

When graphing inequalities, follow these steps:

• Identify the boundary line: This line is found by reinterpreting the inequality as an equation (e.g., replacing \(\geq\) or \(\leq\)) with equality.
• Determine line type: If the inequality involves equality (\(\geq\) or \(\leq\)), use a solid line to represent the boundary. Otherwise, use a dashed line.
• Shade the solution region: Decide which side of the boundary line contains the solutions. For \(\leq\) or \(<\), shade below the line. For \(\geq\) or \(<\), shade above the line.

The intersection of all shaded regions represents the set of points that solves the entire system. Hence, mastering graphing inequalities can help in solving complex problems involving multiple inequalities.

Linear Equations

A linear equation is a mathematical statement that describes a straight-line relationship between two variables. These equations are vital as they represent constant rates of change, able to model everyday situations like business profits or physics problems.

The general form of a linear equation is \(Ax + By = C\), but it's often most useful in slope-intercept form or standard form. In a Cartesian coordinate system, these equations are visualized as straight lines. Each pair of \(x, y\) coordinates that satisfies the equation will lie on this line.

Some key characteristics of linear equations include:

• Constant Rate of Change: The slope represents how much \(y\) changes for a change in \(x\).
• Y-Intercept: The point where the line crosses the y-axis, represented by \(b\) in slope-intercept form.
• Graphical Representation: Intersecting lines can solve systems of equations, often representing solutions through their crossing points.

Understanding linear equations helps in analyzing trends and predictive modeling in numerous fields.

Slope-Intercept Form

The slope-intercept form is a specific way of expressing linear equations that simplifies graphing and understanding straight lines. This form is written as \(y = mx + b\), where \(m\) is the slope of the line, and \(b\) is the y-intercept.

Using this form has several advantages:

• Immediate Clarity: With \(m\) and \(b\) explicitly stated, the line's steepness and position are easily identified.
• Easy to Graph: Start at the y-intercept (\

Slope-Intercept Form

The slope-intercept form is a specific way of expressing linear equations that simplifies graphing and understanding straight lines. This form is written as \(y = mx + b\), where \(m\) is the slope of the line, and \(b\) is the y-intercept.

Using this form has several advantages:

• Immediate Clarity: With \(m\) and \(b\) explicitly stated, the line's steepness and position are easily identified.
• Easy to Graph: Begin plotting at the y-intercept \(b\) on the y-axis. Then, use the slope \(m\) to determine the rise over run to mark another point.
• Straightforward Transformations: Convert standard form or point-slope form into slope-intercept easily by isolating \(y\).

When solving systems of inequalities, slope-intercept form assists in clearly visualizing each inequality's boundary line, critical for accurately shading the solution region. Understanding this form provides a solid foundation for comprehending linear relationships in equations.