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Use graphing technology to graph the solution region of each inequality and then sketch each graph. Adjust the graphing window as needed to show meaningful information.
Here are three situations. There are two questions about each situation. For each question that you work on:
a. Write an inequality to describe the constraints. Specify what each variable represents.
b. Use graphing technology to graph the inequality. Sketch the solution region on the coordinate plane and label the axes.
c. Name one solution to the inequality and explain what it represents in that situation.
d. Answer the question about the situation.
Club Donations
A school environmental club is raising money during the year to donate in support of two causes: a national organization lobbying for clean air and a local non-profit working to restore a stream that runs past the school. They plan to donate a maximum of $600 by the end of the year, some toward clean air and some toward stream restoration.
If the club donates \$200 to stream restoration, what can you say about the amount they donate to clean air?
When the restoration of the stream is complete, the name of any person or group that donates \$500 or more will be added to a commemorative plaque. The national organization lobbying for clean air does not offer any donation perks.
If the club donates enough to get its name on the plaque, what can you say about the amount it donated to each group?
Concert Tickets
Two kinds of tickets to an outdoor concert were sold: lawn tickets and seat tickets. Fewer than 400 tickets in total were sold.
If you know that exactly 100 lawn tickets were sold, what can you say about the number of seat tickets?
Lawn tickets cost \$30 each, and seat tickets cost \$50 each. The organizers want to make at least $14,000 from ticket sales.
If you know that exactly 200 seat tickets were sold, what can you say about the number of lawn tickets?
Advertising Packages
An advertising agency offers two packages for small businesses that need advertising services. A basic package includes only design services. A premium package includes design and promotion. The agency's goal is to sell at least 60 packages in total.
If the agency sells exactly 45 basic packages, what can you say about the number of premium packages it needs to sell to meet its goal?
The basic advertising package has a value of \$1,000 and the premium package has a value of \$2,500. The goal of the agency is to sell more than $60,000 worth of small-business advertising packages.
If you know that exactly 10 premium packages were sold, what can you say about the number of basic packages that the agency needs to sell to meet its goal?
Suppose we want to find the solution to . We can start by graphing the related equation .
When identifying the solution region, it is important not to assume that the solution will be above the line because of a “>” symbol or below the line because of a “<” symbol.
Instead, test the points on the line and on either side of the line, and see if they are solutions.
Graph of a line, origin O, with grid. X axis from negative 2 to 12 by 2’s. Y axis from negative 6 to 8 by 2’s. Line passes through approximately 1 comma negative 4, 5 comma 0, and 12 comma 7.
For , points on the line and above the line are not solutions to the inequality because the pairs make the inequality false. Points that are below the lines are solutions, so we can shade that lower region.
Graphing technology can help us graph the solution to an inequality in two variables.
Many graphing tools allow us to enter inequalities such as and will show the solution region, as shown here.
Some tools, however, may require the inequalities to be in slope-intercept form or another form before displaying the solution region. Be sure to learn how to use the graphing technology available in your classroom.
Inequality graphed on a coordinate plane. X axis from 0 to 10, by 5’s. Y axis from negative 5 to 10, by 5’s. Dashed line goes through 0 comma negative 5, 5 comma 0, and 10 comma 5. The region below the dashed line is shaded.
Although graphing using technology is efficient, we still need to analyze the graph with care. Here are some things to consider:
Inequality graphed on a coordinate plane. Each axis from negative 2 to 4, by 2’s. Dashed line goes through 3 comma negative 2, 3 point 5 comma negative 1 point 5, and 4 comma negative 1. The region below the dashed line is shaded.