Problem 98 - 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. $$y \geq \frac{2}{3} x-2$$

Step-by-Step Solution

Answer

The inequality \(y \geq \frac{2}{3} x-2\) is graphed as a solid line, and the region above this line is shaded to represent the solutions to the inequality.

Step 1: Recognize the Inequality

First, acknowledge that the inequality is in the form \(y \geq mx + b\), where m is the slope and b is the y-intercept. In this case, \(m = \frac{2}{3}\) and \(b = -2\). That indicates that the line has a slope of \(\frac{2}{3}\) and a y-intercept (where the line crosses the y-axis) at -2.

Step 2: Graph the Line

Plot the y-intercept first, which is at -2 on the y-axis. From the y-intercept, use the slope to find the next point. The slope is \(\frac{2}{3}\), meaning for every 3 units moved to the right (positive direction along x-axis), move 2 units up (positive direction along y-axis). Draw the line across these points. Since it's 'greater than or equal to', the line should be solid, indicating that it includes the points on the line.

Step 3: Shading the Area

Because the inequality is 'greater than', the area above the line should be shaded. This represents all the possible solutions to the inequality, meaning any point in this shaded area would satisfy the inequality \(y \geq \frac{2}{3} x-2\).

Key Concepts

Rectangular Coordinate System

The Rectangular Coordinate System, also known as the Cartesian Coordinate System, is a two-dimensional plane used to graph equations and inequalities. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). The point where these axes intersect is called the origin, labeled as (0,0). Each point on this plane is determined by an ordered pair of numbers \(x, y\), where \x\ is the distance from the y-axis and \y\ is the distance from the x-axis.

When graphing an inequality, you're visualizing all the possible solutions on this plane. By placing equations in this system, you can easily identify relationships and intersections between different lines and curves, which are crucial for solving inequalities in two variables.

Slope and Y-intercept

Understanding the slope and y-intercept is essential for graphing lines and inequalities. The **slope** of a line describes how steep the line is and the direction it goes. Mathematically, the slope \( m \) is the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. For example, in the inequality \( y \geq \frac{2}{3}x - 2 \), the slope is \( \frac{2}{3} \). This means that for every three units you move right along the x-axis, the line moves up two units.

The **y-intercept** is where the line crosses the y-axis. In the same inequality, the y-intercept is \(-2\), indicating the point (0, -2). Together, the slope and y-intercept help you plot a starting point on the graph and determine the line's direction. By connecting these dots, you'll have your line accurately represented on the graph.

Shading Regions

When graphing inequalities, shading regions is key. The region that you shade represents all the solutions to the inequality. If your inequality sign is greater than or equal (\( \geq \)), you'll shade above the line. Conversely, if it’s less than or equal (\( \leq \)), you shade below the line. This visual tool helps you see which points satisfy the inequality.

In our exercise, because the inequality is \( y \geq \frac{2}{3}x - 2 \), the region above the solid line is shaded. This shading shows that every point in the shaded area makes the inequality true. Remember that the line itself is also included in the solution set because of the "equal to" part of the inequality, which is why a solid line is used.