Problem 55 - Precalculus Mathematics for Calculus

Question

To graph the solution of a system of inequalities, we have shaded the solution of each inequality in a different color; the solution of the system is the region where all the shaded parts overlap. Here is a different method: For each inequality, shade the region that does not satisfy the inequality. Explain why the part of the plane that is left unshaded is the solution of the system. Solve the following system by both methods. Which do you prefer? Why? $$\left\\{\begin{aligned} x+2 y &>4 \\ -x+y &<1 \\ x+3 y &<9 \\ x &<3 \end{aligned}\right.$$

Step-by-Step Solution

Answer

Method 1 is preferred for visual clarity; the region where all shaded areas overlap is the solution.

Step 1: Understand the Inequalities

First, rewrite each inequality in a form that makes graphing easier. 1. The inequality \( x + 2y > 4 \) can be rewritten as \( y > -\frac{1}{2}x + 2 \).2. The inequality \( -x + y < 1 \) can be rewritten as \( y < x + 1 \).3. The inequality \( x + 3y < 9 \) can be rewritten as \( y < -\frac{1}{3}x + 3 \).4. The inequality \( x < 3 \) is already in a simple form.

Step 2: Graph the First Inequality

Graph the line \( y = -\frac{1}{2}x + 2 \) and shade above the line because we have \( y > -\frac{1}{2}x + 2 \). This represents the solution to the first inequality.

Step 3: Graph the Second Inequality

Graph the line \( y = x + 1 \) and shade below the line because we have \( y < x + 1 \). This is the solution to the second inequality.

Step 4: Graph the Third Inequality

Graph the line \( y = -\frac{1}{3}x + 3 \) and shade below the line because we have \( y < -\frac{1}{3}x + 3 \). This is the solution to the third inequality.

Step 5: Graph the Fourth Inequality

Graph the vertical line \( x = 3 \) and shade to the left because we have \( x < 3 \). This is the solution to the fourth inequality.

Step 6: Identify the Overlapping Region

The solution to the system of inequalities is the region where all four shaded regions overlap. This region satisfies all the inequalities simultaneously.

Step 7: Alternative Method Explanation

An alternative is to shade regions that do not satisfy each inequality (opposite regions).For the inequalities,1. Shade below the line for \( x + 2y > 4 \) (instead of above).2. Shade above for \( -x + y < 1 \) (instead of below).3. Shade above for \( x + 3y < 9 \) (instead of below).4. Shade right for \( x < 3 \) (instead of left).The region left unshaded will be the overlap of opposite regions—the solution.

Step 8: Compare the Methods

The method of directly graphing and looking for where the shaded regions overlap might be visually more straightforward than shading the complement and identifying the unshaded area. Preference depends on ease of visualization.

Key Concepts

System of Inequalities

A system of inequalities consists of two or more inequalities that are considered simultaneously. These inequalities can be linear, and they form a system that requires each individual inequality to be satisfied. Essentially, you are looking for a set of values for the variables that satisfy all the inequalities in the system:

• Each inequality narrows down the possible values for the variables.
• The goal is to find a common region that satisfies all inequalities.

In the given exercise, the system comprises four inequalities involving two variables, \(x\) and \(y\). Each inequality acts as a condition, limiting where the solution can lie on a graph.These constraints come together to form a solution region that satisfies all the inequalities at once. This is where the magic of the system of inequalities lies—it pinpoints a specific area on the graph where all the conditions are met simultaneously. Understanding the interaction of these constraints is crucial to solving any system of inequalities.

Graphical Method

The graphical method is a way to solve systems of inequalities by plotting them on a coordinate plane. This visual representation helps you see how each inequality constrains the possible solutions:

• First, rewrite each inequality in a format that is easy to graph, typically in slope-intercept form \(y = mx + b\).
• Plot each line on the graph and shade the region that satisfies the inequality.

The intersection where the shaded areas overlap is the solution to the system. For example:

• The inequality \(y > -\frac{1}{2}x + 2\) would require shading above the line \(y = -\frac{1}{2}x + 2\).
• Conversely, for \(y < x + 1\), you would shade below the line \(y = x + 1\).

Graphing is an intuitive approach for understanding systems of inequalities because it visually represents all solutions at once. You can easily identify where multiple constraints intersect, providing insight into which areas satisfy all conditions simultaneously.

Solution Region

The solution region is the area on the graph that is not only shaded by all inequalities but remains once we consider the overlaps. It's where all conditions are true at the same time. Here are a few things to note about the solution region:

• Every point in the solution region satisfies each inequality of the system.
• The shape and size of the solution region depend on the original inequalities.

To locate this region, follow these steps:

• Graph each inequality and shade the viable region.
• Find the overlapping area, the intersection of all shaded regions.

In our example, after graphing each inequality based on specified regions (either above or below the line), the solution area is pinpointed where all shadings merge. If you use an opposite shading strategy, the solution region remains unshaded. Both methods will lead you to the same solution region, demonstrating the consistent nature of inequality systems.