Linear Equations, Inequalities and Systems

Lesson 10

Objective

Identify solutions to systems of inequalities graphically. Write systems of inequalities from graphs and word problems.

Common Core Standards

Core Standards

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  • A.CED.A.3 — Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

  • A.REI.D.12 — Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Foundational Standards

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  • 8.EE.C.8

Criteria for Success

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Problem 1

Some treasure has been buried at a point $${(x,y)}$$ on the grid, where $$x$$ and $$y$$ are whole numbers. 

Here are three clues to help you find the treasure: 
Clue 1: $$x> 2$$
Clue 2: $$x+y< 8$$
Clue 3: $$2y-x\geq 0$$

Which of the following points could be a possible location for the treasure? 

(3,2)     (2,3)     (5,3)     (3,5)     (4,3)     (5,2)

Guiding Questions

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Problem 2

Write a system of linear inequalities that only has the region named as part of the solution set. 

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Problem 3

Mary babysits for $4 per hour. She also works as a tutor for $7 per hour. She is only allowed to work 13 hours per week. She wants to make at least $65. Write and graph a system of inequalities to represent this situation.

Guiding Questions

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Anchor Problems

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Problem 1

Some treasure has been buried at a point $${(x,y)}$$ on the grid, where $$x$$ and $$y$$ are whole numbers. 

Here are three clues to help you find the treasure: 
Clue 1:   $$x> 2$$
Clue 2:   $$x+y< 8$$
Clue 3:   $$2y-x\geq 0$$

Which of the following points could be a possible location for the treasure? 

$${(3,2) }$$     $${(2,3)}$$     $${(5,3)}$$      $${(3,5)}$$     $${(4,3)}$$     $${(5,2)}$$

Guiding Questions

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References

MARS Formative Assessment Lessons for High School Representing Inequalities Graphically

Representing Inequalities Graphically from the Classroom Challenges by the MARS Shell Center team at the University of Nottingham is made available by the Mathematics Assessment Project under the CC BY-NC-ND 3.0 license. Copyright © 2007-2015 Mathematics Assessment Resource Service, University of Nottingham. Accessed Oct. 20, 2017, 4:36 p.m..

Modified by The Match Foundation, Inc.

Problem 2

Write a system of linear inequalities that only has the region named as part of the solution set. 

Guiding Questions

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Problem 3

Mary babysits for $4 per hour. She also works as a tutor for $7 per hour. She is only allowed to work 13 hours per week. She wants to make at least $65. Write and graph a system of inequalities to represent this situation.

Guiding Questions

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Problem Set

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The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.

  • Include problems where there is no solution for the system of inequalities as well as all solutions for the system of inequalities.

Target Task

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Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can hold at most eight people. Additionally, each boat can only carry 1,200 pounds of people and gear for safety reasons. Assume an average an adult weighs 150 pounds and a child weighs 75 pounds. Also assume each group will require 200 pounds of gear plus 10 pounds of gear per person.

  1. Write an inequality that illustrates the weight limit for a group of adults and children on the fishing boat and a second inequality that represents the total number of passengers in the fishing boat. Graph and identify at least one solution to the system of inequalities.
  2. Several groups of people wish to rent a boat. Group 1 has 4 adults and 2 children. Group 2 has 3 adults and 5 children. Group 3 has 8 adults. Which of the groups, if any, can safely rent a boat? What other combinations of adults and children are possible?

References