Problem 69 - 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. $$ y \geq \frac{2}{3} x-2 $$

Step-by-Step Solution

Answer

To graph the inequality \(y \geq \frac{2}{3} x-2\), start by graphing the line \(y = \frac{2}{3}x - 2\). Then, shade the region above the line.

Step 1: Identify the slope and y-intercept

The given inequality is in slope-intercept form \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept. Comparing the given inequality \(y \geq \frac{2}{3} x-2\) to the slope-intercept form, it can be inferred that the slope \(m=\frac{2}{3}\) and the y-intercept \(b=-2\).

Step 2: Graph the line

Use the graphing utility to plot the line \(y = \frac{2}{3} x-2\). To do this, start by marking the y-intercept on the y-axis. This is the point (0, -2). From there, use the slope to find the next point. A slope of \(\frac{2}{3}\) means that for every 3 units horizontal move (or move to the right), the line moves 2 units up. So from the y-intercept (0,-2), move 3 units to the right and 2 units up to mark the next point (3, 0). Draw the line through these two points.

Step 3: Shade the appropriate region

Given the inequality is \(y \geq \frac{2}{3}x - 2\), the solution includes all y-values that are greater than or equal to \(\frac{2}{3}x - 2\). Therefore, the region above the line (including the line because of the 'or equals to' part of the inequality) should be shaded.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a popular way to express a linear equation. In this form, a line is represented as \(y = mx + b\), where \(m\) represents the slope, and \(b\) is the y-intercept. This straightforward format allows us to easily identify how the line behaves in the rectangular coordinate system. Let's break it down further:

• Slope (\(m\)): This is the rate at which \(y\) changes for every unit change in \(x\). For the inequality \(y \geq \frac{2}{3}x - 2\), the slope \(m\) is \(\frac{2}{3}\), meaning that the line rises 2 units for every 3 units it moves to the right.
• Y-intercept (\(b\)): The y-intercept is where the line crosses the y-axis. It is the value of \(y\) when \(x\) is 0. In our example, \(b = -2\), indicating that the line crosses the y-axis at the point \((0, -2)\).

This simplicity of expression makes the slope-intercept form particularly useful for quickly graphing linear equations and their corresponding inequalities.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian plane, is a two-dimensional plane formed by the intersection of a vertical y-axis and a horizontal x-axis. It is the foundation for plotting points and lines, allowing us to graph equations and inequalities.

• Axes: The x-axis runs horizontally, while the y-axis runs vertically. These axes are perpendicular to each other and intersect at the origin, denoted by \((0,0)\).
• Coordinates: Every point in the system is represented by an ordered pair \((x, y)\), where \(x\) is the position on the horizontal axis and \(y\) is the position on the vertical axis.
• Quadrants: The plane is divided into four quadrants. The layout of coordinates within each quadrant helps determine the positions of points relative to the origin.

Understanding the rectangular coordinate system is crucial when graphing equations such as \(y \geq \frac{2}{3}x - 2\), as it tells us where to plot lines and which regions to shade based on the inequality's criteria.

Graphing Utilities

Graphing utilities are powerful tools that can assist students in visualizing mathematical concepts, like inequalities, without drawing them by hand. These digital or calculator-based tools help in accurately plotting equations and shading relevant regions as part of the graphical representation of inequalities.

• Plotting Lines: Using a graphing utility, you can input the equation of a line in slope-intercept form or other formats. The tool will then display the line on the coordinate plane, making it simpler to interpret the math behind it.
• Shading Regions: Most graphing utilities offer options to shade regions on the graph to depict solutions to inequalities, like \(y \geq \frac{2}{3}x - 2\). The shading indicates which part of the plane satisfies the inequality, providing a clear visual cue for solutions.
• Checking Solutions: By inputting different points into the utility, you can easily verify if they lie in the shaded region, thus determining if they are solutions to the inequality.

Graphing utilities not only save time but also reduce the likelihood of errors, allowing students to focus more on understanding the mathematical concepts rather than the mechanics of drawing.