Problem 102 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Based on the solution steps, the inequality \(3x - 2y \geq 6\) is graphed by plotting the line y = 1.5x - 3. Testing the origin point (0,0) and finding it doesn't satisfy the inequality, the region opposite to the test point, i.e., above the line, is shaded.

Step 1: Graph the Line

To graph the line, first transition the inequality into a simpler, linear equation by ignoring the inequality for a while. This gives the equation \(3x - 2y = 6\). Rearranging it in y = mx + b format will give \(y = 1.5x - 3\). Plotting that line on a graph is the first step.

Step 2: Determine which Side to Shade

To ascertain which side of the line to shade, select a test point that is not on the line. A typical choice for a reference point is the origin (0,0) if it doesn't belong to the line. Then substitute these x and y values into the original inequality. If the inequality is true with these values, shade the side of the line that includes the test point. If not, shade the other side.

Step 3: Shade the Correct Region

After the test point decides which side of the line to shade, color in that section. The shading represents all of the points (x, y) that will satisfy the inequality \(3x - 2y \geq 6\). The line itself will also be included because of the '\(\geq\)' sign in the inequality. If the sign was \('> only\'), then the line should be dashed to indicate the values on the line don't satisfy the inequality.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system is a two-dimensional plane divided by two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). Each point in this system can be described by an ordered pair \(x, y\), where 'x' is the horizontal coordinate, and 'y' is the vertical coordinate.

This setup is foundational for graphing various mathematical entities, such as lines, curves, and inequalities. The quadrants, which are the four regions created by the intersecting axes, help us determine the position and signs of \(x, y\) coordinates. For example, in the first quadrant, both coordinates are positive.

Using this system helps visualize relationships between variables, underpinning much of algebra and calculus. It's an essential tool for plotting everything from simple lines to complex functions.

Shading Regions

When graphing inequalities, shading regions is critical to illustrate which parts of the coordinate system satisfy the inequality. By determining which side of a boundary line to shade, we visualize feasible solutions for the inequality.

Steps to Shade:

• Graph the boundary line (use a solid line for inequalities with '≥' or '≤' and a dashed line for '>' or '<').
• Choose a test point not on the line, usually the origin, to check which side satisfies the inequality.
• If the test point satisfies the inequality, shade the region that includes it; otherwise, shade the opposite side.

The shaded area represents all points that could be a solution to the inequality. This visual method makes understanding solutions easier, particularly when dealing with multiple inequalities.

Linear Inequalities

Linear inequalities are expressions involving two variables where one side is not equal to the other, as indicated by symbols like '≥', '≤', '>', or '<'. They create a half-plane on the coordinate system, representing an infinite set of solutions.

To solve a linear inequality graphically:

• Convert the inequality to an equation to find the boundary line.
• Determine its slope \(m\) and y-intercept \(b\) to graph the line in the form of \(y = mx + b\).
• Use the inequality symbol to decide whether the line is solid (inclusive) or dashed (exclusive).

The ability to convert and graph these inequalities helps in understanding constraints in real-world problems, like budgets or physical limitations, where solutions must often stay within specific bounds.

Graphing Utilities

Graphing utilities, like calculators and computer software, simplify the task of plotting lines and shading regions. These tools help visualize mathematical concepts quickly, making them invaluable in education and research.

Using Graphing Utilities:

• Input the equation directly into the utility, which will automatically plot the line.
• Use settings within the graphing tool to shade regions based on inequality constraints.
• Experiment with different test points or adjust the view to understand better how inequalities affect regions.

These utilities often come with manuals or help sections, providing step-by-step guidance on performing tasks such as graphing inequalities. Utilizing these tools can significantly enhance one's understanding and efficiency, especially when dealing with complex problems.