Alg1 Lesson 2.5: Systems of Inequalities

Lesson 2.5: Modeling with

Systems of Inequalities

Obj: 3H, 3D, 2H I can formulate, graph, and interpret
solutions to systems of inequalities.

EQ: How do I formulate and solve systems of
inequalities?

Roles:
Facilitator
Scribe
Resourcer
Includer

Lesson Goals:
● Creative Thinking
● Talk through controversies and conflict
● Recognize and reduce ambiguity
● Encourage thinking based on formulas and prior info
● Help explain ideas to each other
● Own your ideas and work
● Record ideas in your journal
● Answer Questions on Slides
● Follow your team roles

Facilitator

• Make sure that all peers are staying on task.

• Give advice or suggestions to resolve the problem.

• Be sure everyone is able to explain.

Scribe

• Make sure peers organize their results on their own papers.

• Remind peers to use color, arrows, and other math tools to
communicate your mathematics, reasons, and connections.

• Be ready to join the teacher for a huddle.

Resourcer

• Make sure peers are getting the materials needed.

• Make sure that all materials are put away neatly.

• Make sure that peers are logged in to the needed site.

• Help troubleshoot any technology difficulties that may arise.

Includer

• Make sure that all peers are talking about their work.

• Helps keep peers’ voice volume low.

• Encourages everyone to ask questions.

• Communicates conflicts or questions to the teacher.

● Check off tasks & skills on calendar.

● Select skills to work on.

● Work on Deltamath.

Remember to work on the following too…

Poll

Go to a table with people working on the same lesson. What will be your role today?

• Includer

• Facilitator

• Resourcer

• Scribe

• No one is working on this lesson right now/I am working from home.

Part 1: Warming Up with a

System of Inequalities

Graph the following System on Desmos

Desmos Graph with Inequalities

https://www.desmos.com/calculator/3i79phmq1j

This link also contains the
{domain} and {range} of
the solution region.

Multiple Choice

How can you tell if a pair of numbers will be a solution to a system of Inequalities?

• A pair of numbers will be a solution if it is a positive integer.
• A pair of numbers will be a solution if it makes all inequalities in the system true.
• A pair of numbers will be a solution if it is the same number repeated.
• A pair of numbers will be a solution if it makes at least one inequality true.

Multiple Choice

If a pair of numbers is a solution to the system of inequalities, what can you
conclude about the coordinates that correspond to the pair of numbers?

• The coordinates will not satisfy the inequalities.
• The coordinates will usually be in quadrant III.

• The coordinates will always be on the boundary line.

• The coordinates will lie outside the shaded region.
• The coordinates will lie in the shaded region of the graph.

Multiple Choice

Is (4, 2) a solution to the system?

• No, (4, 2) is not a solution and will not lie in the shaded region.
• Yes, (4, 2) is not a solution but will lie in the shaded region.
• No, (4, 2) is a solution but will not lie in the shaded region.
• Yes, (4, 2) is a solution and will lie in the shaded region.

Categorize

Options (7)

Any point in the shaded region

Any point on a dashed line

(2.3, 3.8)

(0, 0)

(−1.5, 2.1)

(0, 4)

(3, -1)

Classify each as a solution or not a solution.

Solution

Not a Solution

Multiple Choice

In general, is it possible to write down all the pairs of numbers that are solutions to a
system of inequalities?

• No

• Yes

Multiple Choice

Is it possible to write down all the pairs of integers that are solutions to the system of inequalities?

• Yes

• No

Write down all pairs
of integers that are
solutions to this
system of
inequalities.

Multiple Choice

How many integer solutions are there to this system of inequalities?

• 30

• 21

• 17

• 11

• 4

Part 2: Writing and Solving

a System of Inequalities

from a Context

A Local Rancher

A local rancher raises steers and hogs for a local livestock
group, and there has been a recent increase in demand. For
every one steer the group purchases, there are at least two
hogs purchased. The rancher wants to expand and raise more
animals to meet this demand. Each steer requires 15 square
yards (yd²) of space in his pen, and each hog pen requires 9 yd²
of space. The rancher calculates that she has space available
for a total of 120 yd² of new pens.

Multiple Select

What are the quantities involved in the scenario?

• steers

• hogs

• fences

• money

• chickens

Drag and Drop

What are some of the important phrases in the problem to write the inequalities? “​

Placeholder

,​​

Placeholder

." and

"

Placeholder

in his pen, and ​

Placeholder

.”

Drag these tiles and drop them in the correct blank above

For every one steer the group purchases, there are

A local rancher raises steers and hogs

The rancher wants to expand

raise more animals to meet this demand

at least two hogs purchased

Each steer requires 15 square yards (yd2) of space

each hog pen requires 9 yd2 of space

Multiple Choice

What kind of numbers would you expect to solve this problem?

• whole numbers
• fractions
• negative numbers
• irrational numbers

Think about this Statement and Use to Answer Questions

For every one steer the group purchases, there
are at least two hogs purchased.

Multiple Choice

Using the information here, if there was one steer purchased by the group, how many hogs would have
been purchased?

• hogs > 2

• hogs < 2

• hogs ≤ 2

• hogs ≥ 2

Multiple Select

Now think about if 2 steers were purchased. According to the statement, how many hogs is a possible amount we should purchase?

• 4

• 3

• 2

• 5

• 8

Multiple Select

What if 5 steers were purchased? According to the statement, how many hogs should we expect to have
been purchased?

• 10

• 25

• 7

• 5

• 8

Multiple Choice

If s represents the total number of steers and h represents the total number of hogs, what could be an inequality that describes the relationship between the number of steers purchased
and the number of hogs purchased?

• h ≥ 2s

• s ≥ 2h

• h ≤ 2s

• s ≤ 2h

Think about this Statement and Use to Answer Questions

Each steer requires 15 square yards (yd²) of space in his pen, and each hog pen requires 9 yd² of space. The rancher
calculates that she has space available for a total of 120 yd² of new pens.

Multiple Choice

How much space does each steer need in its pen? How could we calculate the total amount of space all
the steers need?

• 15 + s
• 15s

• s²
• 9s

Multiple Choice

How much space does each hog need in its pen? How could we calculate the total amount of space all
the hogs need?

• 120/h

• 15h

• h²

• 9h

Drag and Drop

The total space needed for both types of pens can be expressed by ​

Placeholder

. The rancher has ​120 yd² for pens. She could have exactly ​120 yd² of new pens, but she cannot go ​

Placeholder

​120 yd². The amount of space must be
​

Placeholder

120 yd².

Drag these tiles and drop them in the correct blank above

15s + 9h

over

less than or equal to

greater than or equal to

less than

greater than

under

15s - 9h

Multiple Choice

What inequality could we write to express the relationship between the total amount of space the steer and hog pens will take up and the amount of space the rancher has available?

• 15s + 9h ≤ 120
• 10s + 5h ≤ 100
• 15s + 9h ≥ 120
• 20s + 15h ≤ 150

Multiple Choice

What is the minimum number of steers and hogs that the rancher could raise?

• 0 steers and 0 hogs

• 3 steers and 1 hog
• 0 steers and 2 hogs
• 2 steers and 0 hogs

Categorize

Options (4)

2 steer pens and 4 hog pens

3 steer pens and 4 hog pens

1.5 steer pens and 7.25 hog pens

the same number of steer and hog pens

Can Buy

Cannot Buy

Part 3: Finding an Optimal

Solution

Material Costs

The rancher knows that it costs $165
to build each steer pen and $90 for
each hog pen.

Dropdown

To calculate the total cost of building steer and hog pens, we should
​

Select answer

165 by the number of steers, s, and ​

Select answer

90 by the number of hogs, h.
Then we can add the quantities: ​

Select answer

.

Fill out the table
for each purchase
combination.

Part 4: Summary & Practice

Handout 2.5: Modeling with Systems of Inequalities

Random Question of the Day Time

https://wheelofnames.com/4ke-epz We’ll spin the
wheel as a class and spend a minute or so
discussing our answers.