Problem 73 - 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. $$ 3 x-2 y \geq 6 $$

Step-by-Step Solution

Answer

Rearrange the equation to \(y \leq \frac{3}{2}x -3 \), draw the line \(y = \frac{3}{2}x -3 \), and then shade the section above the line.

Step 1: Rearrange the inequality

Rearrange \( 3x - 2y \geq 6 \) by isolating y. Subtract \(3x\) on both sides to get \( -2y \geq -3x + 6 \), then divide by \(-2\) to get \(y \leq \frac{3}{2}x -3\)

Step 2: Identify the Boundary Line

The boundary line for the given inequality will be the line \(y = \frac{3}{2}x -3 \). Plot this line on the graph using the graphing utility.

Step 3: Choose a Test Point

Choose a test point not on the line to identify which area to shade. A commonly chosen test point is the origin \((0,0)\). Substitute these values into the original inequality. If it holds true, then the region towards the origin will be shaded, if not, the region away from the origin will be shaded.

Step 4: Shade the Region

Upon substituting \((0,0)\) into our original inequality, it wouldn't satisfy the equation. So, we shade away from the origin. Therefore, the upper area from the line will be shaded, representing the solution to the inequality.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also known as Cartesian coordinate system, is a two-dimensional plane consisting of two perpendicular lines, called axes. The horizontal axis is generally referred to as the x-axis, while the vertical axis is called the y-axis. Each point in this system can be represented by an ordered pair \( (x, y) \), indicating its position in terms of distance from the origin, where both axes intersect. Graphical representation of functions, including inequalities, is a vital application of this system.

When graphing inequalities, you'll often start by sketching the related boundary line, which typically represents where the inequality holds as an equality. For a thorough understanding, imagine every single point on the plane: a particular point either satisfies the inequality, thereby being part of the solution region, or does not. The graph thus serves as a visual summary of all possible solutions.

Inequality in Two Variables

An inequality in two variables, like \( 3x - 2y \geq 6 \), defines a set of points on the rectangular coordinate system that makes the inequality true. Unlike an equation that corresponds to a single line, an inequality represents a region. For linear inequalities in two variables, this region is always a half-plane, bounded by the line of the associated linear equation (which is obtained when you replace the inequality symbol with an equality).

In our example, solving \( 3x - 2y \geq 6 \) for \(y\) gives us the boundary line \(y = \frac{3}{2}x -3\). However, every point below or above this line must be tested to see if it satisfies the original inequality, determining which side of the line will ultimately be shaded to represent the solution set.

Using Graphing Utilities

Graphing utilities, such as graphing calculators or software, are incredibly handy for visualizing and solving inequalities. To graph an inequality using these tools, you typically enter the equation of the boundary line. Then, you can use the utility's function to shade the appropriate region of the graph—in our case, representing all the points that satisfy \( 3x - 2y \geq 6 \).

These utilities simplify the process by allowing you to quickly test regions without plotting individual points. They provide the big picture at a glance, showcasing the solutions in a graphical format. It's always good practice to know how to use the shading feature of your specific graphing tool, as this method is not only efficient but also reduces the potential for manual errors.

Boundary Line

The boundary line plays a pivotal role in graphing two-variable inequalities. It represents the edge of the solution set where the inequality is true as an equation. For a linear inequality like \(y \leq \frac{3}{2}x -3\), the boundary line will be \(y = \frac{3}{2}x - 3\). It's vital to determine whether this line is part of the solution set (solid line) or not (dashed line).

In our example, the inequality \(3x - 2y \geq 6\) will result in a solid boundary line because the equal part of \(\geq\) indicates that the points on the line are included in the solution. Typically, you'll choose a test point to see which side of the boundary line is included in the solution set. For \geq and >, if the test point satisfies the inequality, you shade the region that includes the test point; otherwise, you shade the opposite side.