4.5 Graphing a System of Inequalities

Posljednje ažuriranje over 1 year ago

Napomena autora:

Instrukcije

Intro-Warm-Up

Learn It

Practice It

Apply It

Level 3: Bonus

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.

4.5 Graphing a System of Inequalities

Posljednje ažuriranje over 1 year ago

Napomena autora:

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

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

Instrukcije

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

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

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.

State two different combinations of hours that Ben can work at each job to meet all of the criteria of this situation.

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.

null

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.

null

3. Find overlap

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

null

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)

Solve the system of linear inequalities by graphing in Desmos. Then, state two solutions.

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.

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.

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.

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)

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

State a point that falls in the new solution region.

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.

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.

Graph the system of inequalities on the coordinate plane.

State 3 points that are in the solution space and what they mean in the context of the theater department’s budgeting situation.

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

What does each variable represent about the barbeque?

What does the second inequality tell us about the situation?

Graph the system of inequalities using the grid below

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

Level 3: Bonus

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.

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

4.5 Graphing a System of Inequalities

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.

4.5 Graphing a System of Inequalities

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

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

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

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

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

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)

Level 1

Graph the system of inequalities on the coordinate plane.

What does the second inequality tell us about the situation?

Graph the system of inequalities using the grid below

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

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.

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

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

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

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

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

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

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)

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

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

Level 1

Graph the system of inequalities on the coordinate plane.

What does the second inequality tell us about the situation?

Graph the system of inequalities using the grid below

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

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

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

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

4.5 Graphing a System of Inequalities

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.

4.5 Graphing a System of Inequalities

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

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

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

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

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

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)

Level 1

Graph the system of inequalities on the coordinate plane.

What does the second inequality tell us about the situation?

Graph the system of inequalities using the grid below

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

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.

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 equationsDefine all variablesCreate a graph that shows that (4, 7) is a solution

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

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

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

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

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

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)

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*

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

Level 1

Graph the system of inequalities on the coordinate plane.

What does the second inequality tell us about the situation?

Graph the system of inequalities using the grid below

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

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 equationsDefine all variablesCreate a graph that shows that (4, 7) is a solution

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*

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

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)

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

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)

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

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

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