Problem 54 - A Graphical Approach to College Algebra

Question

Graph the solution set of each system of inequalities by hand. $$\begin{array}{c} x+y \leq 4 \\ x-y \leq 5 \\ 4 x+y \leq-4 \end{array}$$

Step-by-Step Solution

Answer

The solution set is the intersecting shaded region bounded by all inequalities.

Step 1: Understanding the Inequalities

We have a system of three inequalities: \( x + y \leq 4 \), \( x - y \leq 5 \), and \( 4x + y \leq -4 \). These inequalities define regions on a coordinate plane.

Step 2: Graph the Line for \( x + y = 4 \)

First, graph the line \( x + y = 4 \). Find two points by setting first \( x = 0 \) and \( y = 0 \) to see where the line crosses the axes. For \( x = 0 \), \( y = 4 \). For \( y = 0 \), \( x = 4 \). Plot these points and draw the line. Shade below this line because the inequality is \( \leq \).

Step 3: Graph the Line for \( x - y = 5 \)

Graph this line by finding where it crosses the axes. Set \( x = 0 \) to get \( y = -5 \), and \( y = 0 \) to get \( x = 5 \). Connect these points and draw the line. Shade below the line because the inequality is \( \leq \).

Step 4: Graph the Line for \( 4x + y = -4 \)

To graph this line, find intercepts by setting \( x = 0 \) to find \( y = -4 \) and \( y = 0 \) to solve for \( x = -1 \). Plot \( (0, -4) \) and \( (-1, 0) \), draw the line, and shade below it, as the inequality is \( \leq \).

Step 5: Identify the Overlapping Region

The solution is the region where all three shaded areas overlap. Identify this region on the graph, which satisfies all the given inequalities simultaneously.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we plot points, lines, and curves to visualize mathematical concepts, such as inequalities. It's made up of two axes: the horizontal axis is known as the x-axis, and the vertical axis is called the y-axis. These axes intersect at the origin, noted as the point (0,0).

To plot a point on this plane, we need two numbers, an x-coordinate and a y-coordinate, together forming an ordered pair (x, y). The x-coordinate tells us how far to move horizontally from the origin, while the y-coordinate tells us how far to move vertically. For example, the point (3,2) lies 3 units to the right of the origin and 2 units upwards.

When dealing with inequalities like those in the given problem, we graph each inequality's boundary line on the coordinate plane. These lines divide the plane into different regions. Our ultimate goal is to find and represent the specific area that meets all inequality conditions.

Solution Set

A solution set consists of all possible points on the coordinate plane that simultaneously satisfy all the given inequalities in a system. In our exercise, we are dealing with three inequalities, and the solution set is where the shaded regions from these inequalities overlap.

The process generally involves:

• Graphing each inequality's boundary line on the plane.
• Shading the correct region as indicated by the inequality sign.

Once all inequalities are graphed, observe the areas where the shaded regions intersect. This common intersecting area represents the solution set, visualizing all the points that satisfy every inequality in the system.

Finding the solution set visually on the graph makes it easier to comprehend the relationship and restrictions imposed by the inequalities.

Inequality Shading

Inequality shading involves filling or coloring the area on one side of an inequality's boundary line to indicate all the points that make the inequality true. Each inequality in the problem has a boundary line, like the line for the equation version, such as \(x + y = 4\). However, unlike a standard line equation, inequalities specify a half of the coordinate plane to be shaded.

Here's how we determine which side to shade:

• Graph the corresponding equality line first.
• Pick a test point not on the line (often the origin (0,0) is used if it’s not on the line).
• Substitute this point into the inequality.
• If the inequality holds true, shade the side of the line that includes the test point; otherwise, shade the opposite side.

Remember to shade the area that fulfills the inequality (for \(\leq\) or \(\geq\)), this tells us all the solutions for that particular inequality. In our problem, doing this for all the inequalities helps us visually understand the constraints each inequality places on the solution set.