Problem 25 - Intermediate Algebra

Question

Graph the solution set of each system of inequalities on a rectangular coordinate system. $$\left\\{\begin{array}{l}y \geq x \\\y \leq \frac{1}{3} x+1 \\\x>-3 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution set is the region above \( y=x \), below \( y=\frac{1}{3}x+1 \), and right of \( x=-3 \).

Step 1: Plotting the Line of First Inequality

The first inequality is \( y \geq x \). First, plot the line \( y = x \), which is a straight line through the origin with a slope of 1. Since the inequality is \( y \geq x \), shade the region above the line.

Step 2: Plotting the Line of Second Inequality

The second inequality is \( y \leq \frac{1}{3}x + 1 \). Plot the line \( y = \frac{1}{3}x + 1 \). This line has a y-intercept of 1 and a slope of \( \frac{1}{3} \). The line is solid, and you should shade the region below it since it is a \( \leq \) inequality.

Step 3: Drawing the Vertical Line for Third Inequality

The third inequality is \( x > -3 \). Plot a vertical dashed line at \( x = -3 \) to indicate that the area to the right (where \( x > -3 \)) is included in the solution set.

Step 4: Determining the Solution Region

Combine the information from all inequalities. The solution region is the area where all shaded regions intersect, which lies above the line \( y = x \), below the line \( y = \frac{1}{3}x + 1 \), and to the right of the line \( x = -3 \).

Key Concepts

Understanding Graphing Inequalities

Graphing inequalities involves representing linear inequalities on a coordinate plane to find the combination of variables that satisfy all given conditions. When you graph inequalities, you use lines to separate the coordinate plane into regions. Each region corresponds to solutions that satisfy or do not satisfy the inequality.

• Solid lines represent inequalities that include the value, as seen in the case of \( \leq \) or \( \geq \).
• Dashed lines represent inequalities that do not include the value, indicated by \( < \) or \( > \).
• Shaded areas show where the solutions exist for the inequality.

In the given problem, each inequality represents a different condition that our solutions must meet, and we graph these on the coordinate plane to find the common solution region.

The Role of the Coordinate System

A coordinate system is a crucial tool to represent different mathematical equations visually. It typically includes two axes: the x-axis (horizontal) and the y-axis (vertical), which intersect at a point called the origin, denoted as (0, 0).
The coordinate plane allows us to plot points and lines described by equations or inequalities. By using the coordinate system, we can graph linear lines and regions of inequalities to easily observe and identify where solutions overlap. This is especially useful for systems of inequalities, like in our original exercise, where interpreting intersections provides the desired solution set.

Linear Inequalities Explained

Linear inequalities are similar to linear equations but instead use inequality symbols like \( \leq \), \( \geq \), \(<\), or \(>\) to represent a range of possible solutions. They can be visualized as a line (solid or dashed) on a coordinate plane, with solutions falling on one side of the line.
A key aspect of linear inequalities is understanding how they divide the coordinate plane into two half-planes. Depending on the inequality symbol, we choose one half-plane as the solution region.
For example:

• The inequality \( y \geq x \) means that the solution includes points where y is greater than or equal to x, indicating shading above the line \( y = x \).
• With \( y \leq \frac{1}{3} x + 1 \), the shading will be below the line because it includes points less than or equal to the line.

Identifying where these shaded regions overlap results in the solution region for the system of inequalities.