Problem 102 - College Algebra

Question

Graphing utilities can be used to shade regions in the rectangular coordinate system, thereby graphing an inequality in two variables Read the section of the user's manual for your graphing utility that describes how to shade a region. Then use your graphing utility to graph the inequalities in Exercises 97-102. $$3 x-2 y \geq 6$$

Step-by-Step Solution

Answer

The graph of the inequality \(3x - 2y \geq 6\) is a solid line represented by \(y = 1.5x -3\) and the shaded region above this line.

Step 1: Set Inequality to Equation Form

Rewrite the inequality \(3x - 2y \geq 6\) into equation form to get the line boundary. This gives \(3x - 2y = 6\) or equivalently \(y = 1.5x -3\). This is the equation we will plot.

Step 2: Graph the Line

Plot the line \(y = 1.5x -3\) on the coordinate system using your graphing utility. Since the inequality is 'greater than or equal to' type, the line should be solid, indicating that the points on the line are included in the solution.

Step 3: Determine the Region to Shade

Since the original inequality is \(3x - 2y \geq 6\), the solution includes all the points making this inequality true. Choose a test point not on the line, typically the origin \((0,0)\) is a good choice if it doesn't lie on the line. Substituting \(x=0, y=0\) into the inequality, we get \(-2(0) - 3(0) \geq 6\), or \(0 \geq 6\) which is false. This means we shade the side of the line that does not include the origin.

Step 4: Shade the Correct Region

Shade the region above the line on the graphical representation in the graphing utility. All the points in this shaded region, including the points on the line, are solutions to the inequality \(3x - 2y \geq 6\).

Key Concepts

Coordinate System

Understanding the coordinate system is fundamental for graphing inequalities. It's where we visually represent numerical relationships. The coordinate system, commonly known as the Cartesian plane, consists of two perpendicular axes: the horizontal axis (x-axis) and the vertical axis (y-axis). These axes intersect at a point called the origin, designated as \(0,0\).

Every point on the plane is represented by a pair of numbers \(x, y\), which correspond to positions along the x-axis and y-axis, respectively. The coordinate system is divided into four quarters, called quadrants, which help determine the sign of \(x\) and \(y\) values. By using the coordinate system, we can easily plot equations and inequalities to visualize the solutions.

Graphing Utility

A graphing utility is a tool that helps us plot functions and inequalities. This can be a software program, an app, or a dedicated graphing calculator. When using a graphing utility to represent an inequality, it provides an efficient way to see the region of the coordinate plane that satisfies the inequality.

For our exercise, the graphing utility assists us by automating the plotting process and shading the correct region, which is extremely helpful for visualizing the relation between \(x\) and \(y\). Graphing utilities often come with a variety of features such as zooming, tracing points, and setting up different window settings, which allows for a thorough exploration of the graph.

Inequality in Two Variables

An inequality in two variables like \(3x - 2y \geq 6\) represents a relationship between two variables where instead of having an exact equality, we have a range of possible solutions. Rather than plotting a single line as with an equation, you would represent an inequality by shading a region of the coordinate system.

This highlights all the pairs of \(x\) and \(y\) that make the inequality true. The orientation of the inequality sign \(\geq\ or \leq\) determines which side of the line to shade. If the inequality includes 'equal to' (as in \(\geq\) or \(\leq\)), the boundary line itself is part of the solution set, and thus it is drawn as a solid line. If the inequality is strict (\(>\) or \(<\)), the line is dashed to indicate that points on the line are not solutions.

Linear Equation Graphing

When we talk about linear equation graphing, we refer to the process of drawing the graph of a linear equation, which forms a straight line. Every point on this line is a solution to the equation. The standard form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The slope tells us how steep the line is, and the y-intercept tells us where the line crosses the y-axis.

In our example, \(y = 1.5x - 3\) is the linear equation derived from the original inequality. Graphing this equation means plotting the line on the coordinate system, which in turn acts as the boundary for the inequality solution. In our case, since we're dealing with \(\geq\), all points on the line and above it (to infinity in the y-direction) are included in the solution set.