Question

Use a graphing calculator or a computer to graph the system of inequalities. Give the coordinates of each vertex of the solution region. \(x+2 y \geq 0\) \(5 x-2 y \leq 0\) \(-x+y \leq 3\)

Short answer

It's necessary to use a graphing calculator or a computer software to accurately determine the vertices, but the method involves plotting the inequalities and identifying the points of intersection.

Step-by-step solution

Plot Inequality 1

Plot the inequality \(x + 2y \geq 0\) on a coordinate system. The area above and including the line represents the solutions to the inequality

Plot Inequality 2

Next, plot the inequality \(5x - 2y \leq 0\) on the same coordinate system. The area beneath and including the line represents the solutions to this inequality.

Plot Inequality 3

Finally, plot the inequality \(-x + y \leq 3\) on the same coordinate system. The area beneath and including this line is the solution region for this inequality.

Identify the Solution Region

Now, the solution region for the entire system of inequalities is the area where all individual solution regions overlap. This region is a polygon.

Identify the Vertices

The vertices of the solution region are the points where two or more of the boundary lines (the lines for each inequality) intersect. These are the coordinates that need to be identified.

Key concepts

Graphing Calculator

When tackling problems that involve graphing systems of inequalities, a graphing calculator emerges as an invaluable tool. Not simply a device for basic arithmetic or algebraic calculations, a graphing calculator allows you to visually represent complex mathematical problems on a digital coordinate grid. It transforms equations and inequalities into lines, parabolas, or other shapes.

For students, this visual representation is key to understanding where solutions might lie. By inputting the inequality formulas, the graphing calculator will shade the regions that satisfy each inequality. This shading helps to identify the solution region, especially when multiple inequalities must be considered simultaneously.

Exercise Improvement Advice

• Firstly, practice entering multiple inequalities into the calculator to see how each region is shaded.
• Use the zoom and trace features to precisely find the intersection points which represent the vertices of the solution region.

When using the graphing calculator for the exercise provided, you would enter each inequality one at a time, and the tool would graphically display the regions, making it easier to pinpoint the solution region and its vertices.

Solution Region

The solution region is the heart of solving a system of inequalities. It is the physical area on a graph where the solutions to all inequalities in the system overlap. Think of it as the 'common ground' shared by all inequalities involved. Each inequality contributes to narrowing down the possibilities until only the solution region remains.

Illustrated on a graph, it often appears as shaded space, being either polygonal or curvilinear, depending on the inequalities' nature. The boundaries of this region may be lines (as in linear inequalities) or curves (as in non-linear inequalities), and they might include the lines themselves (if the inequality is inclusive) or not (if the inequality is strict).

Exercise Improvement Advice

• While graphing, use different colors or shading patterns for each inequality to avoid confusion.
• Check for the solution region where all the shadings overlap, and make sure you understand why it represents the solution to the system.

The exercise's solution region, a polygon, is defined by the vertices at the intersection of the inequalities' boundary lines.

Coordinate System

A coordinate system is a framework that allows you to locate points on a graph. The most common system used in algebra is the Cartesian coordinate system, which consists of two axes perpendicular to each other: the horizontal x-axis and the vertical y-axis. Where these axes intersect is called the origin, the (0,0) point from which all other points are measured.

When graphing inequalities, each point on the coordinate plane is checked against the inequalities to determine if it is part of the solution set. The beauty of the Cartesian system lies in its simplicity and universality; it's used everywhere from basic algebra to advanced calculus.

Exercise Improvement Advice

• Always label your axes and scale them appropriately to best fit all the lines and intersections within your graph.
• Remember that the origin is your point of reference; use it to plot inequalities and identify solution regions accurately.

By applying the coordinate system effectively, students can precisely plot inequalities and identify where they intersect, leading to the discovery of the solution region's vertices in the exercise.

Inequality Graphing

Inequality graphing is a technique used to solve systems of inequalities visually. Each inequality can be treated as a boundary, with one side of the boundary containing the solutions. In graphing these boundaries, you essentially partition the coordinate plane into regions where the inequalities are valid.

A solid line is used on the graph when the inequality includes the boundary (meaning it's either \(\geq\) or \(\leq\)), and a dashed line for strict inequalities (\(>\) or \(<\)). The shading technique is crucial; it highlights the side of the boundary where the inequality is true.

Exercise Improvement Advice

• Start by plotting the equal part of the inequality to form the boundary lines.
• Determine the side of the boundary to shade by choosing a test point, not on the line, to see if it satisfies the inequality.

Following these steps will help depict the solution regions for each inequality, and where they overlap is where the solution to the entire system lies—as demonstrated in the provided exercise.