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4.5 Graphing a System of Inequalities

Last updated 10 months ago
26 questions
Note from the author:
OBJECTIVES & STANDARDS
Math Objectives
  • Graph systems of linear inequalities
  • State solutions to the system using the solution region on the graph
  • Understand special cases of systems of linear inequalities
Common Core Math Standards
  • Link to all CCSS Math
  • CCSS.PRACTICE.MP1
  • CCSS.HSA.REI.D.12
  • CCSS.HSF.LE.B.5
Personal Finance Objectives
  • Work with a budget given restraints
National Standards for Personal Financial Education
Spending
  • 1b: Develop a budget to allocate current income to necessary and desired spending, including estimates for both fixed and variable expenses.
DISTRIBUTION & PLANNING
Distribute to students
  • Student Activity Packet
  • Application Problems
OBJECTIVES & STANDARDS
Math Objectives
  • Graph systems of linear inequalities
  • State solutions to the system using the solution region on the graph
  • Understand special cases of systems of linear inequalities
Common Core Math Standards
  • Link to all CCSS Math
  • CCSS.PRACTICE.MP1
  • CCSS.HSA.REI.D.12
  • CCSS.HSF.LE.B.5
Personal Finance Objectives
  • Work with a budget given restraints
National Standards for Personal Financial Education
Spending
  • 1b: Develop a budget to allocate current income to necessary and desired spending, including estimates for both fixed and variable expenses.
DISTRIBUTION & PLANNING
Distribute to students
  • Student Activity Packet
  • Application Problems
Intro-Warm-Up
INTRO: How Many Hours Should Ben Work?
Read the scenario below and use it to answer the questions.
Ben is working two jobs: Delivering pizzas at $12 an hour and working at a grocery store making $15 an hour. Between the two jobs, he can work a maximum of 20 hours per week and needs to make at least $250 to cover his expenses.
2
State two different combinations of hours that Ben can work at each job to meet all of the criteria of this situation._______ _______
1

Can you think of a way to summarize ALL possible solutions without having to make a list?

Learn It
What is a System of Inequalities?
As you can see from the Intro question, there are some situations that have multiple different answers and it can be challenging to summarize ALL of them using words.  Systems of Inequalities give us the ability to summarize all of the possible solutions using a graph!
A system of linear inequalities is a set of two or more linear inequalities that are graphed on the same coordinate plane.
The solution of a system of linear inequalities is the region of the graph where all shaded regions overlap.  All points in the region make both inequalities true.

Review the example problem to solve the following system of linear inequalities by graphing.
y > -2x + 4
y < x - 2

1. Graph the first inequality
Graph the border line, determine dashed (< or >) or solid line (< or >), then use a test point to determine the solution region.


2. Graph the second inequality
Graph the border line, determine dashed (< or >) or solid line (< or >), then use a test point to determine the solution region.


3. Find overlap
Find the area of the graph that is shaded in for both inequalities.


4. Identify solutions.
Solutions are points that fall in the shaded region for BOTH inequalities. Remember points on dashed lines are NOT solutions but points on solid lines ARE solutions.
Some example solutions:
(4, 0)
(6, 2)
(7, -1)
(8, 4)
3
Solve the system of linear inequalities by graphing in Desmos. Then, state two solutions.
_______ _______
3
Solve the system of linear inequalities by graphing. Then, state two solutions._______ _______
3
Solve the system of linear inequalities by graphing. Then, state two solutions._______ _______
3
Solve the system of linear inequalities by graphing. Then, state two solutions._______ _______
Practice It
For each system of inequalities, graph to show the solution region and state one solution to the system.
1

For each system of inequalities, graph to show the solution region and state one solution to the system.

1

For each system of inequalities, graph to show the solution region and state one solution to the system.

1

For each system of inequalities, graph to show the solution region and state one solution to the system.

1

For each system of inequalities, graph to show the solution region and state one solution to the system.

Practice It
Systems of Inequalities Applications
Let’s revisit Ben from the Intro. Here’s the scenario again:
Ben is working two jobs: delivering pizzas at $12 an hour and working at a grocery store making $15 an hour. Between the two jobs, he can work a maximum of 20 hours per week and needs to make at least $250 to cover his expenses.
1
Write two inequalities that represent Ben’s situation where x is the number of hours worked delivering pizzas and y is the number of hours worked at the grocery store._______ _______
1

Use this Desmos Graph to enter your equations and view the solution region. Tip: To enter < in Desmos, just type <=

(screen shot your answer into the drawing)

1
Using your graph, state 4 points that are in the overlapping solution region and what they mean to Ben._______ _______ _______ _______
1

If Ben’s expenses changed to at least $1000 instead of $250, how would your equations change? Make those changes in Desmos. (You will need to zoom out to see both graphs)


*screen shot the graphs in the drawing*

1
State a point that falls in the new solution region._______
1

What does your answer to question 5 mean about Ben covering $1000 in expenses in a week?

Apply It
APPLICATION: Graphing a System of Budget Inequalities

Level 1

Budgeting for the Big Show
Your high school theater department is setting their budget for their annual spring musical. They know that their theater can hold a maximum of 1,300 people for the show. They plan to sell tickets for $9 in advance and $12 on the day of the show. They know that they must make at least $12,500 to cover the costs of their budgeted expenses.
1
Write a system of linear inequalities that represents this situation. Let x represent the number of tickets sold in advance and y represent the number of tickets sold on the day of the show._______ _______
1

Graph the system of inequalities on the coordinate plane.

1
State 3 points that are in the solution space and what they mean in the context of the theater department’s budgeting situation._______ _______ _______
2
State 1 point that is a solution to one of the inequalities but not the other._______
What does this point represent in the theater department’s budgeting situation?_______
Sammi is organizing an end of summer barbeque. She needs to make at least 50 sandwiches and has a budget of $30. She goes to the grocery store to get buns. The grocery store sells hamburger buns in packages of 8 and hotdog buns in packages of 6.
The following system of equations represents Sammi’s barbecue planning:
8x + 6y > 50
3.5x + 2.25y < 30
2
What does each variable represent about the barbeque?
x=_______
y=_______
1

What does the second inequality tell us about the situation?

1

Graph the system of inequalities using the grid below

1

State 3 solutions and what they mean in the context of Sammi’s barbecue.

Level 3: Bonus
0

Write a system of linear inequalities that has the point (4, 7) as a solution. At least one of your equations should have a slope not equal to zero.

0

Write a word problem that tells a story about your system of inequalities. Be sure to:
  • Describe all parts of the story that contribute to your equations
  • Define all variables
  • Create a graph that shows that (4, 7) is a solution