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Name: _______________________________________________ Engage NY Lessons 12-15

(6th Grade Lesson 34): Writing and Graphing Inequalities in Real-World Problems

Inequality Symbol Graphing Point

At Least >

More Than >

At Most <

Less Than <

Example 1

Statement Inequality Graph

a. Caleb has at least $5.

b. Tarek has more than $5.

c. Vanessa has at most $5.

d. Li Chen has less than $5.

Example 2

Kelly works for Quick Oil Change. If customers have to wait longer than 20 minutes for the oil change

the company does not charge for the service. The fastest oil change that Kelly has ever done took 6

minutes. Show the possible customer wait times in which the company charges the customer. Write

an inequality equation and graph on a number line.

a) What is the fastest oil change that Kelly has done? ____________________

b) If customers have to wait longer than _______ minutes, then the oil change is free.

c) What is the range of time between the shortest and longest amount of time?

d) Write 2 inequalities to represent this range. Use “x” as the variable.

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e) Graph the inequalities from (d).

f) How do you think we could write this inequality using the range of values, but only one

variable?

g) So, instead of overlapping the arrows, graph the inequality with 2 points, but one line.

Example 3

Gurnaz has been mowing lawns to save money for a concert. Gurnaz will need to work for at least six

hours to save enough money but he must work less than 16 hours this week. Write an inequality to

represent this situation, and then graph the solution.

Example 4

Each type of fish thrives in a specific range of temperatures. The optimum temperatures for sharks

range from 18 degrees Celsius to 22 degrees Celsius. Write an inequality that represents the

temperatures where sharks will NOT thrive. Write an inequality to represent this situation and graph it

on a number line.

Problem Set

Write an inequality to represent each situation. Then graph the solution.

1. Blayton is at most 2 meters above sea level.

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2. Edith must read for a minimum of 20 minutes.

3. Travis milks his cows each morning. He has never gotten less than 3 gallons of milk however he

always gets less than 9 gallons of milk.

4. Rita can make 8 cakes for a bakery each day. So far she has orders for more than 32 cakes.

Right now, Rita needs more than four days to make all 32 cakes.

5. Rita must have all the orders placed right now done in 7 days or less. How will this change your

inequality and your graph?

Write an inequality equation, solve with 7 steps of algebra and graph the solution.

6. Kasey has been mowing lawns to save up money for a concert. He earns $15 per hour and

needs at least $90 to go to the concert. How many hours should he mow?

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7. Rachel can make 8 cakes for a bakery each day. So far she has orders for more than 32 cakes.

How many days will it take her to complete the orders?

8. Ranger saves $70 each week. He needs to save at least $2,800 to go on a trip to Europe. How

many weeks will he need to save?

9. Clara has less than $75. She wants to buy 3 pairs of shoes. What price shoes can Clara afford if all

the shoes are the same price?

10. A gym charges $25 per month plus $4 extra to swim in the pool for an hour. If a member only has

$45 to spend each month, at most how many hours can the member swim?

Problem Set : Write and graph an inequality for each problem.

11. At least 13.

12. Less than 7.

13. Chad will need at least 24 minutes to complete the 5K race. However, he wants to finish in under

30 minutes.

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14. Eva saves $60 each week. Since she needs to save at least $2,400 to go on a trip to Europe, she

will need to save for at least 40 weeks.

15. Clara has $100. She wants to buy 4 pairs of the same pants. Due to tax, Clara can afford pants

that are less than $25.

16. A gym charges $30 per month plus $4 extra to swim in the pool for an hour. Because a member

has just $50 to spend at the gym each month, the member can swim 5 hours at most.

Lesson 12: Properties of Inequalities

There are four stations. Each station 2 dice.

1. Roll each die, recording the numbers under the first and third columns. Students are to write

an inequality symbol that makes the statement true. Repeat this four times to complete the

four rows in the table.

2. Perform the operation indicated at the station (adding or subtracting a number, writing

opposites, multiplying or dividing by a number), writing a new inequality statement.

3. Determine if the inequality symbol is preserved or reversed when the operation is performed.

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Station 1 or 5: Add or Subtract a Number to Both Sides of the

Inequality

Die 1 Inequality Die 2 Operation New Inequality

Inequality Symbol

Preserved or

Reversed?

−3 < 5 Add 2 −3 + 2 < 5 + 2

−1 < 7 Preserved

Add −3

Subtract 2

Subtract

−1

Add 1

**Examine the results. Make a statement about what you notice, and justify it with

evidence.

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Station 2 or 6: Multiply each term by −𝟏

Die 1 Inequality Die 2 Operation New Inequality

Inequality Symbol

Preserved or

Reversed?

−3 < 4 Multiply by −1

(−1)(−3) < (−1)(4)

3 < −4

3 > −4

Reversed

Multiply by −1

Multiply by −1

Multiply by −1

Multiply by −1

**Examine the results. Make a statement about what you notice and justify it with

evidence.

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Station 3 or 7: Multiply or Divide Both Sides of the Inequality by a

Positive Number

Die 1 Inequality Die 2 Operation New Inequality

Inequality Symbol

Preserved or

Reversed?

−𝟐 > −𝟒 Multiply by 𝟏

𝟐

(−𝟐) ( 𝟏

𝟐) > (−𝟒) (

𝟏

𝟐)

−𝟏 > −𝟐 Preserved

Multiply by 𝟐

Divide by 𝟐

Divide by 𝟏

𝟐

Multiply by 𝟑

**Examine the results. Make a statement about what you notice, and justify it with

evidence.

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Station 4 or 8: Multiply or Divide Both Sides of the Inequality by a Negative

Number

Die 1 Inequality Die 2 Operation New Inequality

Inequality Symbol

Preserved or

Reversed?

𝟑 > −𝟐 Multiply by −𝟐 𝟑(−𝟐) > (−𝟐)(−𝟐)

−𝟔 < 𝟒 Reversed

Multiply by −𝟑

Divide by −𝟐

Divide by −𝟏

𝟐

Multiply by −𝟏

𝟐

**Examine the results. Make a statement about what you notice and justify it with

evidence.

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When you finish the stations, complete this table.

Complete the following chart using the given inequality, and determine an operation in which the

inequality symbol is preserved and an operation in which the inequality symbol is reversed. Explain

why this occurs.

Inequality

Operation and New

Inequality Which

Preserves the

Inequality Symbol

Operation and New

Inequality Which

Reverses the

Inequality Symbol

Explanation

7 < 8

7 < 8

Add 4 to both sides

7 + 4 < 8 + 4

11 < 12

7 < 8

Multiply Both Sides by -4

7 (-4) > 8 (-4)

-28 > -32

Adding a number to both sides

of an inequality PRESERVES the

inequality symbol.

Multiplying a negative number

to both sides of an inequality

REVERSES the inequality symbol.

2 < 5

−4 > −6

−1 ≤ 2

−2 + (−3) < −3 − 1

Lesson Summary

When both sides of an inequality are added or subtracted by a number, the inequality symbol stays

the same and the inequality symbol is said to be preserved.

When both sides of an inequality are multiplied or divided by a positive number, the inequality symbol

stays the same and the inequality symbol is said to be preserved.

When both sides of an inequality are multiplied or divided by a negative number, the inequality

symbol switches from < to > or from > to <. The inequality symbol is reversed.

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Lesson 12 Homework Problem Set

1. For each problem, use the properties of inequalities to write a true inequality statement.

Two integers are −2 and −5.

a. Write a true inequality statement.

b. Subtract −2 from each side of the inequality. Write a true inequality statement.

c. Multiply each number by −3. Write a true inequality statement.

2. In science class, Melinda and Owen are experimenting with solids that disintegrate after an initial

reaction. Melinda’s sample has a mass of 155 grams, and Owen’s sample has a mass of 180

grams. After one minute, Melinda’s sample lost one gram and Owen’s lost three grams. For each

of the next ten minutes, Melinda’s sample lost one gram per minute and Owen’s lost three grams

per minute.

a. Write an inequality comparing the two sample’s masses after one minute.

b. Write an inequality comparing the two masses after four minutes.

c. Explain why the inequality symbols were preserved.

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3. On a recent vacation to the Caribbean, Kay and Tony wanted to explore the ocean elements.

One day they went in a submarine 150 feet below sea level. The second day they went scuba

diving 75 feet below sea level.

a. Write an inequality comparing the submarine’s elevation and the scuba diving elevation.

b. If they only were able to go one-fifth of the capable elevations, write a new inequality to

show the elevations they actually achieved.

c. Was the inequality symbol preserved or reversed? Explain.

4. If 𝑎 is a negative integer, then which of the number sentences below is true? If the number

sentence is not true, give a reason.

a. 5 + 𝑎 < 5

b. 5 + 𝑎 > 5

c. 5 − 𝑎 > 5

d. 5 − 𝑎 < 5

e. 5𝑎 < 5

f. 5𝑎 > 5

g. 5 + 𝑎 > 𝑎

h. 5 + 𝑎 < 𝑎

i. 5 − 𝑎 > 𝑎

j. 5 − 𝑎 < 𝑎

k. 5𝑎 > 𝑎

l. 5𝑎 < 𝑎

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Lesson 13: Inequalities

Classwork

Opening Exercise: Writing Inequality Statements

Tarik is trying to save $265.49 to buy a new tablet. Right now he has $40 and can save $38 a

week from his allowance.

Write and evaluate an expression to represent the amount of money saved after:

2 Weeks:

𝟒𝟎 + 𝟑𝟖(𝟐)

𝟒𝟎 + 𝟕𝟔

𝟏𝟏𝟔

3 Weeks:

4 Weeks:

5 Weeks:

6 Weeks:

7 Weeks:

8 Weeks:

When will Tarik have enough money to buy the tablet?

Write an inequality that would generalize this problem for money being saved for 𝑤 weeks.

Why is it possible to have more than one solution?

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What is the minimum amount of money that Tarik needs to buy the tablet? _________________

What inequality would demonstrate this? ____________________________

Interpret the meaning of the 38 and 40 in the inequality.

Exercise 1:

1. Connor went to the county fair with a $22.50 in his pocket. He bought a hot dog and drink for

$3.75, and then wanted to spend the rest of his money on ride tickets which cost $1.25 each.

a. Write an inequality to represent the total spent where 𝑟 is the number of tickets

purchased.

b. Connor wants to use this inequality to determine whether he can purchase 10 tickets.

Use substitution to show whether or not he will have enough money.

c. What is the total maximum number of tickets he can buy based upon the given

information? Use the 7 steps of algebra to solve.

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Exercise 2

Write and solve an inequality statement to represent the following problem:

On a particular airline, checked bags can weigh no more than 𝟓𝟎 pounds. Sally packed 𝟑𝟐 pounds

of clothes and five identical gifts in a suitcase that weigh 𝟖 pounds. Write an inequality to represent

this situation. Solve using the 7 steps of algebra.

Let x = the weight of one gift.

Problem Set

1. Shaggy earned $7.55 per hour plus an additional $100 in tips waiting tables on Saturday. He

earned at least $160 in all. Write an inequality and find the minimum number of hours, to the

nearest hour, Shaggy worked on Saturday.

2. Match each problem to the inequality that models it. One choice will be used twice.

_ ____ The sum of three times a number and −4 is greater than 17. a. 3𝑥 + −4 ≥ 17

___ __ The sum of three times a number and −4 is less than 17. b. 3𝑥 + −4 < 17

___ __ The sum of three times a number and −4 is at most 17. c. 3𝑥 + −4 > 17

__ ___ The sum of three times a number and −4 is no more than 17. d. 3𝑥 + −4 ≤ 17

__ ___ The sum of three times a number and −4 is at least 17.

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3. If 𝑥 represents a positive integer, find the solutions to the following inequalities.

a. 𝑥 < 7

b. 𝑥 − 15 < 20

c. 𝑥 + 3 ≤ 15

d. −𝑥 > 2

e. 10 − 𝑥 > 2

f. −𝑥 ≥ 2

g. 𝑥

3< 2

h. −𝑥

3> 2

i. 3 −𝑥

4> 2

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4. Recall that the symbol ≠ means "not equal to." If 𝑥 represents a positive integer, state

whether each of the following statements is true or false.

a. 𝑥 > 0

b. 𝑥 < 0

c. 𝑥 > −5

d. 𝑥 > 1

e. 𝑥 ≥ 1

f. 𝑥 ≠ 0

g. 𝑥 ≠ −1

h. 𝑥 ≠ 5

5. Twice the smaller of two consecutive integers increased by the larger integer is at least 25. Model

the problem with an inequality, and determine which of the given values 7, 8, and/or 9 are

solutions. Then find the smallest number that will make the inequality true.

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6. The length of a rectangular fenced enclosure is 12 feet more than the width. If Farmer Dan has 100 feet of

fencing, write an inequality to find the dimensions of the rectangle with the largest perimeter that can be

created using 100 feet of fencing.

What are the dimensions of the rectangle with the largest perimeter? What is the area enclosed

by this rectangle

7. At most, Kyle can spend $50 on sandwiches and chips for a picnic. He already bought chips for

$6 and will buy sandwiches that cost $4.50 each. Write and solve an inequality to show how many

sandwiches he can buy. Show your work and interpret your solution.

Lesson 14: Solving Inequalities

Opening Exercise

You are the owner of the biggest and newest rollercoaster called the ‘Gentle Giant’. The

rollercoaster costs $6 to ride. The operator of the ride must pay $200 per day for the ride rental and

$65 per day for a safety inspection. If you want to make a profit of at least $1000 each day, what is

the minimum number of people that must ride the rollercoaster to make that profit?

What is the revenue?

What are the daily expenses?

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Write an inequality that can be used to find the minimum number of people, 𝒑, that must ride

the rollercoaster each day to make the daily profit. Solve using the 7 steps of algebra.

Interpret the solution.

What if the expenses were charged for a whole day versus a half day? How would

that change the inequality and answer?

Example 1

A youth summer camp has budgeted $2000 for the campers to attend the carnival. The cost

for each camper is $17.95, which includes general admission to the carnival and 2 meals.

The youth summer camp must also pay $250 for the chaperones to attend the carnival and

$350 for transportation to and from the carnival. What is the greatest amount of campers

that can attend the carnival if the camp must stay within their budgeted amount? Write an

equation and use the 7 steps of algebra to solve this problem.

Let c = number of campers to attend the carnival.

Recall profit is the revenue

(money received) less the

expenses (money spent).

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Example 2

The carnival owner pays the owner of an exotic animal exhibit $650 for the entire time the

exhibit is displayed. The owner of the exhibit has no other expenses except for a daily

insurance cost. If the owner of the animal exhibit wants to make more than $500 in profits for

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2 days, what is the greatest daily insurance rate he can afford to pay? Write an

equation and use the 7 steps of algebra to solve this problem.

Let i = daily insurance cost.

Write an equivalent inequality clearing the decimals.

Why do we multiply by 10 to clear the decimals and not 100?

Example 3

There are several vendors at the carnival who sell products and also advertise their

businesses. Shane works for a recreational company that sells ATVs, dirt bikes, snowmobiles,

and motorcycles. His boss paid him $500 for working all of the days at the carnival plus 5%

commission on all of the sales made at the carnival. What was the minimum amount of sales

Shane needed to sell if he earned more than $1,500?

Let s = sales in dollars, made during the carnival.

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How can we write an equivalent inequality containing only integer coefficients and constant

terms? Write the equivalent inequality.

Solve the new inequality using the 7 steps of algebra.

Lesson Summary

The goal to solving inequalities is to use If-then moves to make 0s and 1s to get the inequality into the form

𝑥 > a number or 𝑥 < a number. Adding or subtracting opposites will make 0s. According to the If-then

move, a number that is added or subtracted to each side of an inequality does not change the solution

of the inequality. Multiplying and dividing numbers makes 1s. A positive number that is multiplied or

divided to each side of an inequality does not change the solution of the inequality. However,

multiplying or dividing each side of an inequality by a negative number does reverse the inequality sign.

Given inequalities containing decimals, equivalent inequalities can be created which have only integer

coefficients and constant terms by repeatedly multiplying every term by ten until all coefficients and

constant terms are integers.

Given inequalities containing fractions, equivalent inequalities can be created which have only integer

coefficients and constant terms by multiplying every term by the least common multiple of the values in

the denominators.

Problem Set : Write an inequality for each problem. When asked to solve, use the 7 steps of

algebra. Don’t forget to define the variable with the statement, let __ =.

1. As a salesperson, Jonathan is paid $50 per week plus 3% of the total amount he sells. This week,

he wants to earn at least $100. Write an inequality with integer coefficients for the total sales

needed and describe what the solution represents.

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2. Systolic blood pressure is the higher number in a blood pressure reading. It is measured as the

heart muscle contracts. Heather was with her grandfather when he had his blood pressure

checked. The nurse told him that the upper limit of his systolic blood pressure is equal to half his

age increased by 110.

a. 𝑎 is the age in years and 𝑝 is the systolic blood pressure in mmHg (milliliters of Mercury). Write

an inequality to represent this situation.

b. Heather’s grandfather is 76 years old. What is “normal” for his systolic blood pressure?

3. Traci collects donations for a dance marathon. One group of sponsors will donate a total of $6 for

each hour she dances. Another group of sponsors will donate $75 no matter how long she

dances. What number of hours, to the nearest minute, should Traci dance if she wants to raise at

least $1,000?

4. Jack’s age is three years more than twice his younger brother’s, Jimmy’s, age. If the sum of

their ages is at most 18, find the greatest age that Jimmy could be.

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5. Brenda has $500 in her bank account. Every week she withdraws $40 for miscellaneous

expenses. How many weeks can she withdraw the money if she wants to maintain a balance

of a least $200?

6. A scooter travels 10 miles per hour faster than an electric bicycle. The scooter traveled for 3

hours, and the bicycle traveled for 51

2 hours. All together, the scooter and bicycle travelled no

more than 285 miles. Find the maximum speed of each.

Lesson 15: Graphing Solutions to Inequalities

Exercise 1

Two identical cars need to fit into a small garage. The opening is 23 feet 6 inches wide, and there

must be at least 3 feet 6 inches of clearance between the cars and between the edges of the

garage. How wide can the cars be?

Draw a diagram of this problem. Graph all the points on the number line below.

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What inequality could be used to solve this problem?

Example 1

A local car dealership is trying to sell all of the cars that are on the lot. Currently, it has 525 cars on

the lot, and the general manager estimates that they will consistently sell 50 cars per week. Estimate

how many weeks it will take for the number of cars on the lot to be less than 75.

Write an inequality that can be used to find the number of 𝑤 full weeks. Since 𝑤 is the

number of full or complete weeks, when 𝑤 = 1 means at the end of week 1.

Solve using the 7 steps of algebra and graph your results.

Interpret the solution in the context of the problem.

Verify the solution.

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Exercise 2

The cost of renting a car is $25 per day plus a one time fee of $75.50 for insurance. How

many days can the car be rented if the total cost is to be no more than $525?

Write an inequality to model the situation. Let x = ___________________________________

Solve using the 7 steps of algebra and graph the inequality.

Interpret the solution in the context of the problem.

a) Why do we use rays when graphing the solutions of an inequality on a number line?

b) When graphing the solution of an inequality on a number line, how do you determine what

type of circle, open or closed, to use?

c) When graphing the solution of an inequality on a number line, how do you determine the

direction of the arrow?

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Problem Set:

For each problem, write an inequality, define the variable, solve using the 7 steps of algebra, graph

the inequality, and interpret the solution within the context of the problem.

1. Mrs. Smith decides to buy three sweaters and a pair of jeans. She has $120 in her wallet. If

the price of the jeans is $35, what is the highest possible price of a sweater?

2. The members of the Select Chorus agree to buy at least 250 tickets for an outside concert.

They buy 20 fewer lawn tickets than balcony tickets. What is the least number of balcony

tickets bought?

3. Samuel needs $29 to download some songs and movies on his iPod. His mother agrees to

pay him $6 an hour for raking leaves in addition to his $5 weekly allowance. What is the

minimum number of hours Samuel must work in one week to have enough money to

purchase the songs and movies?

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4. The junior-high art club sells candles for a fundraiser. The first week of the fundraiser the club sells 7

cases of candles. Each case contains 40 candles. The goal is to sell at least 13 cases. During the

second week of the fundraiser, the club meets its goal. Write, solve, and graph an inequality that

can be used to find the possible number of candles sold the second week.

5. Ben has agreed to play less video games and spend more time studying. He has agreed

to play less than 10 hours of video games each week. On Monday through Thursday, he

plays video games for a total of 51

2 hours. For the remaining 3 days, he plays video

games for the same amount of time each day. Find t, the amount of time he plays video

games, for each of the 3 days. Graph your solution.

6. Gary’s contract states that he must work more than 20 hours per week. The graph below

represents the number of hours he can work in a week.

𝟏𝟕 𝟏𝟖 𝟏𝟗 𝟐𝟎 𝟐𝟏 𝟐𝟐

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a. Write an algebraic inequality that representing the number of hours, ℎ, Gary can

work in a week.

b. Gary gets paid $15.50 per hour in addition to a weekly salary of $50. This week he

wants to earn more than $400. Write an inequality to represent this situation.

c. Solve using the 7 steps of algebra and graph the solution form part (b).

7. A bank account has $650 in it. Every week, Sally withdraws $50 to pay for her dog sitter.

What is the maximum number of weeks that Sally can withdraw the money so there is at

least $75 remaining in the account? Write and solve an inequality to find the solution and

graph the solution on a number line.

8. On a cruise ship, there are two options for an internet connection. The first option is a fee

of $5 plus an additional $0.25 per minute. The second option $50 for an unlimited number

of minutes. For how many minutes, 𝑚, is the first option cheaper than the second option?

Graph the solution.

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9. The length of a rectangle is 100 centimeters, and its perimeter is greater than 400

centimeters. Henry writes an inequality and graphs the solution below to find the width of

the rectangle. Is he correct? If yes, write and solve the inequality to represent the

problem and graph. If no, explain the error(s) Henry made.

Sprint Results: Write your score & place for each sprint.

1. Solving Equations Sprint #1: Score _______________ Place _______________

2. Inequalities Sprint #1: Score _______________ Place _______________

3. Inequalities Sprint #2: Score _______________ Place _______________

4. Inequalities Sprint #3: Score _______________ Place _______________

Puzzle

Cut the puzzle into 16 smaller squares. Mix up the pieces. Put the pieces together to form a 4 × 4

square. When pieces are joined, the problem on one side must be attached to the answer on the

other. All problems on the top, bottom, right, and left must line up to the correct graph of the

solution.

𝟗𝟔 𝟗𝟕 𝟗𝟖 𝟗𝟗 𝟏𝟎𝟐 𝟏𝟎𝟑 𝟏𝟎𝟒 𝟏𝟎𝟎 𝟏𝟎𝟏

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Notes:

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