Graphing Inequalities (Vertical and Horizontal Boundaries)

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SWBAT graph inequalities with vertical and horizontal boundaries.

Big Idea

This lesson is actually a continuation of the Two Variable Inequality lesson. Again, the topic of inequalities is a great one for modeling with real world problems.

Accessing Prior Knowledge

15 minutes

When students enter the classroom, the following word problem is projected on the whiteboard. Tell them to work this problem out with their “elbow” partner.

"You are going shopping for school supplies. Pencils cost 25 cents, and pens cost

50 cents.You have budgeted 5 dollars for both. What are the different ways you

could buy pencils and pens while staying within your budget?"

Resource: Beginning_Problem_VertHorizIneq

Allow the students about 5 minutes to discuss the problem. I inform them that what is being sought is an inequality representing the problem, along with its graph. Allow them to use graph paper, their calculator, or the calculator if they have access.  

Once finished, I call on someone and ask…. “ok, how did you and your partner develop your ideas, and what inequality did you come up with?”

They should have concluded with something like 0.25x + 0.50y ≤ 5. Ask other students if they came up with something different.

A “sometimes” error:

Be on the lookout for students who forget we are dealing with money and write the equation as:

25x + 50y ≤ 5

I then open and ask the student to come up and graph the inequality on the screen for all to see. Ask the following questions while the graph is up:

  • Where are the solutions to our “pencil-pen” problem? (Students should indicate that the solutions are in the triangular region constrained by the line and both axes)  
  • Are all the points in this triangular region a solutions to our problem? (If students have not said this already,  they should indicate that only those whole number coordinates in this region are solutions, because one cannot buy fractions of pens or pencils)
  • State a few valid possible examples of purchases. (Correct possibilities are (10, 5),  (8, 2), (2, 7)…)
  • What does the entire shaded region represent? (Students should say that it represents all solutions to the inequality, even those that don’t represent valid solutions to the problem)
  • State a solution to the inequality that is not a solution to our pen and pencil problem. (sample answers are (-2, -4), (0, -10)
  • State a set of ordered pairs which represent a purchase above our budget. (Any point above the line in the un-shaded region is correct.)

New Info/Application

30 minutes

Tell students that some inequalities are quite easy to graph and find the shaded region for. These are the ones whose boundaries are vertical or horizontal.

With the graph of .25x + .5y ≤ 5 projected on the board, tell the students to think about the following scenario…

Suppose that you had  to buy exactly 10 pencils, how many pens could you be able to buy to stay within the budget of 5 bucks?

Ask them to solve this manually. (Students should substitute x for 10 in the inequality and solve for y, obtaining y ≤ 5. They should say that you can  purchase 5, 4, 3, 2, 1 or 0 pens)

I then use the desmos graph already projected to demonstrate the inequality y ≤ 5 resulting from substituting 10 for x.

The following video narrative poses this scenario as well as one where the number of pens to buy must be four….leading to the inequality x ≤ 12, which they are asked to solve manually as well. 

Video Narrative



15 minutes

After the work and the desmos demonstation, or any video demonstration for that matter, I like to ask the students to stand and stretch a bit….get blood circulating ….and then I tell them to take a few moments to think about how they can literally summarize the objective of this lesson in a paragraph of  no more than three sentences. I ask each student to write their paragraph on a piece of paper. Once they’ve done so, I ask a few students to share what they wrote. I like to write a good paragraph on the board for all to copy into their notebooks.

I proceed to end the lesson using the 6 question resource (Closing _6Questions.docx). I like to do this orally. Most get this pretty well by now, and you will most certainly get a choir of correct answers.


Extension and Homework

Extension Problem:

Suppose you have another pen and pencil problem. The prices, 25 cents for pencils and 50 cents for pens, and the budget, 5 bucks, remain the same. Yet you must purchase at least 3 pencils and at least 2 pens. Graph this scenario on a coordinate plane.

You should develop three inequalities, including our original  .25x +.50y ≤ 5

and plot 3 different graphs on the same coordinate plane. 

Homework: HOMEWORK.docx