Problem 50 - A Graphical Approach to College Algebra

Question

Graph the solution set of each system of inequalities by hand. $$\begin{array}{r} x \geq 0 \\ x+y \leq 4 \\ 2 x+y \leq 5 \end{array}$$

Step-by-Step Solution

Answer

The solution set is the region bounded by the lines \( x = 0 \), \( y = -x + 4 \), and \( y = -2x + 5 \).

Step 1: Interpret each inequality

The given system of inequalities consists of three inequalities: \( x \geq 0 \), \( x + y \leq 4 \), and \( 2x + y \leq 5 \). These inequalities describe the region in the coordinate plane where each condition is satisfied. We need to graph each inequality on the coordinate plane.

Step 2: Graph the boundary line for \( x \geq 0 \)

The inequality \( x \geq 0 \) indicates that the solution set includes the region to the right of the \( y \)-axis, including the axis itself. This is a vertical line on the \( y \)-axis and the area to the right of it.

Step 3: Graph the boundary line for \( x + y \leq 4 \)

The inequality \( x + y \leq 4 \) can be rewritten as \( y \leq -x + 4 \). To graph this, plot the line \( y = -x + 4 \) by finding intercepts: when \( x = 0 \), \( y = 4 \), and when \( y = 0 \), \( x = 4 \). The solution includes the region below this line.

Step 4: Graph the boundary line for \( 2x + y \leq 5 \)

Rewrite \( 2x + y \leq 5 \) as \( y \leq -2x + 5 \). Graph the line \( y = -2x + 5 \) by calculating intercepts: when \( x = 0 \), \( y = 5 \), and when \( y = 0 \), \( x = 2.5 \). The area below this line is the solution for this inequality.

Step 5: Identify the solution set

The solution of the system of inequalities is the region where the shaded areas of all inequalities overlap. This is a convex polygon bounded by the lines \( y = -x + 4 \), \( y = -2x + 5 \), and the line \( x = 0 \). Check boundary points to ensure they satisfy all inequalities.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we plot points, lines, and shapes based on coordinates. It's also known as the Cartesian plane.

The plane consists of two axes: the x-axis (horizontal) and the y-axis (vertical). They intersect at a point called the origin (0, 0). This system allows us to represent any point on the plane using a pair of numbers (x, y).

• The x-coordinate indicates the horizontal position of a point. Positive x-values are to the right of the origin, and negative x-values are to the left.
• The y-coordinate represents the vertical position. Positive y-values are above the origin, while negative y-values are below.

Plotting on the coordinate plane helps in visualizing mathematical concepts such as equations, functions, and inequalities. It is especially useful in graphing inequalities because it allows us to see which regions satisfy the inequalities.

System of Inequalities

A system of inequalities consists of two or more inequalities that share the same variables. The solution to such a system is the set of values that satisfy all inequalities in it simultaneously.

Consider the system: - \(x \geq 0\) - \(x + y \leq 4\) - \(2x + y \leq 5\)Each inequality has its own region of solutions, and when these overlapping regions are plotted on the coordinate plane, they form the solution set for the system.

• Intersection of Solution Regions: The solution to the system is the intersection of all the solution regions. This means the shaded area where all conditions are true is the solution set.
• Graphing: To graph, draw each inequality. Shade the region that represents the solution for each inequality. The area where all regions overlap is your solution set.

Linear Inequalities

Linear inequalities resemble linear equations but use inequality symbols like \(<\), \(>\), \(\leq\), and \(\geq\) instead of an equal sign. These symbols indicate that the solutions include a range of values rather than a single solution.

Each linear inequality divides the coordinate plane into two regions:

• Boundary Line: This is the line you would plot if the inequality was an equation. For example, \(y = -x + 4\) is the boundary line for \(x + y \leq 4\).
• Shaded Region: This represents all the values that satisfy the inequality. For \(x + y \leq 4\), shade the region below the line \(y = -x + 4\) because it satisfies the condition.

To check if a point is within the solution set of an inequality, substitute its coordinates into the inequality. If the inequality holds true, then the point is a part of the solution set. This method is crucial when identifying the overlapping region that solves a system of inequalities.