Problem 71 - A Graphical Approach to College Algebra

Question

Use the shading capabilities of your graphing calculator to graph each inequality or system of inequalities.$$\begin{aligned} &x+y \geq 2\\\ &x+y \leq 6 \end{aligned}$$

Step-by-Step Solution

Answer

Shade the region between the lines \( x + y = 2 \) and \( x + y = 6 \).

Step 1: Interpret the Inequalities

The given inequalities are: \( x + y \geq 2 \) and \( x + y \leq 6 \). They suggest that the regions above or on the line \( x + y = 2 \) and below or on the line \( x + y = 6 \) should be shaded.

Step 2: Graph the Line for \( x + y = 2 \)

To graph the line \( x + y = 2 \), rewrite it as \( y = -x + 2 \). Plot the y-intercept at \( (0, 2) \) and use the slope \(-1\) to find another point, like \( (2, 0) \). Draw the line through these points. Since \( x + y \geq 2 \), shade the region above this line.

Step 3: Graph the Line for \( x + y = 6 \)

For the line \( x + y = 6 \), rewrite it as \( y = -x + 6 \). Plot the y-intercept at \( (0, 6) \) and use the slope \(-1\) to find another point, like \( (6, 0) \). Draw the line through these points. Since \( x + y \leq 6 \), shade the region below this line.

Step 4: Identify the Solution Region

The solution to the system of inequalities is the area where the shaded regions from Step 2 and Step 3 overlap. This region will be between the lines \( x + y = 2 \) and \( x + y = 6 \). Use the shading feature of your graphing calculator to emphasize this overlapping region.

Key Concepts

Systems of Inequalities

When dealing with systems of inequalities, we're looking at two or more inequalities that share the same set of variables. In this exercise, we have two linear inequalities in two variables: \( x + y \geq 2 \) and \( x + y \leq 6 \). These inequalities are grouped together because they share the same variables, \( x \) and \( y \). The goal is to find a region on a graph that satisfies both conditions simultaneously.
To solve a system of inequalities, we graph each inequality and determine the solution region where all inequalities are true.

• Each inequality forms a line on the graph.
• The inequalities indicate which side of the line should be shaded.
• The overlap of shaded regions from all inequalities represents the solution region.

Understanding systems of inequalities helps us grasp more complex mathematical concepts, particularly where multiple conditions must be satisfied.

Graphing Calculator

A graphing calculator is a valuable tool for visualizing mathematical concepts, especially when dealing with complex graphs. These calculators allow you to quickly and accurately plot lines, curves, and shaded regions.
When graphing inequalities, you can use the shading feature to highlight specific areas that meet the conditions set out by the inequalities. Here are some tips on how to utilize a graphing calculator for this task:

• Input the inequality equations as functions. This might require rearranging them to standard form, such as \( y = -x + c \).
• Use the calculator to plot each line individually.
• Select the shading option to visually interpret which side of the line fulfills the inequality condition.
• Look for the intersection of the shaded areas to identify the solution region.

Using a graphing calculator can simplify the problem-solving process and provide a clearer understanding of solution regions.

Linear Equations

Linear equations are equations in which the highest power of the variable is one. They form straight lines when graphed.
In this exercise, we have two linear equations derived from the inequalities: \( x + y = 2 \) and \( x + y = 6 \). These equations help us determine the boundaries of our solution region.

• The equation can be rewritten in slope-intercept form, \( y = mx + b \), to make graphing easier.
• The slope \( m \) helps you determine how steep the line is, while \( b \) is the y-intercept where the line crosses the y-axis.

Graphing these linear equations will provide a visual guide to finding where inequalities overlap, indicating all possible solutions.

Solution Region

The solution region is the area on the graph where all conditions of a system of inequalities are satisfied concurrently. This region is typically found by observing where the shaded areas from each inequality overlap.
Here's how to identify this crucial region:

• Graph each inequality individually and create a shaded area for each one.
• Look for the part of the graph where these shaded areas intersect. This intersection represents the solution region where both conditions are true.
• Ensure that the boundaries of this region are clearly defined by the lines \( x + y = 2 \) and \( x + y = 6 \).

In this example, the solution region lies between the two parallel lines where both inequalities hold true, showcasing the practical function of systems of inequalities in defining feasible regions.