We included **HMH Into Math Grade 7 Answer Key PDF** **Module 8 Lesson 2 Write Two-Step Inequalities for Situations **to make students experts in learning maths.

## HMH Into Math Grade 7 Module 8 Lesson 2 Answer Key Write Two-Step Inequalities for Situations

I Can write two-step inequalities for real-world scenarios and determine whether given values are solutions.

**Spark Your Learning**

A small business owner is planning to upgrade the computer system. A new system will cost $1,200, but the total cost can be more with installation. The owner has saved $300 and will continue to save $30 from profits each day for the new system. How many days will it take the business owner to save enough money to purchase the new computer system?

Select the inequality that represents this situation. Describe what each part of the inequality represents.

300 + 30x ≥ 1.200

300 + 30x < 1,200

300 + 30x ≤ 1,200

300 + 30x > 1,200

Answer:

300 + 30x ≥ 1.200

Explanation:

Owner has saved $300 and will continue to save $30 from profits each day for the new system.

300 + 30x ≥ 1.200

**Turn and Talk** Describe a situation that could be represented by changing the direction of the inequality symbol in the inequality above. Explain.

Answer:

300 + 30x ≤ 1,200

Explanation:

A small Business owner is planning to buy stock for their stop.

His budget is $1200 only.

He has already bought 10 items each $30.

**Build Understanding**

As you have learned, a one-step inequality involves one operation. For example, x + 1 > 3 is a one-step inequality. In this example, 4, 40, and 4,000,000 are all solutions because they make the inequality true when substituted for x, but 1 is not a solution because it does not make the inequality true when substituted for x.

A two-step inequality involves two operations.

Question 1.

Caitlyn wants to buy sheets of trumpet music at the price shown. She has only $25, and she first needs to pay back $5 to her friend. How many sheets can Caitlyn buy?

A. Describe how this problem represents an inequality situation.

Answer:

She has no more than $25 to spend

B. Describe the part of this situation that is variable.

Answer:

Number of sheets of trumpet music is a variable.

C. What are all the costs that Caitlyn’s money will cover?

Answer:

6 sheets of trumpet music and $5 to her friend.

D. Write an expression to represent all the costs that Caitlyn’s money will cover.

Answer:

3x < 20

E. Write an inequality for this situation.

Answer:

x < 20/3

F. State a possible solution for this inequality.

Answer:

x < 6

x < 20/3 < 6

G. State a value for the variable that would NOT be a possible solution for this inequality.

Answer:

20/3 = 6.66

Explanation:

Caitlyn wants to buy sheets of trumpet music at the price shown is a variable.

She has only $25, and she first needs to pay back $5 to her friend.

Cost of trumpet music per sheet is $3

20/3 = 6.66

x < 20/3 < 6

**Turn and Talk** If Caitlyn wanted more than $4 left over, how would you change the inequality? Explain.

Answer:

3x + 4 < 20 + 4

Explanation:

Cost of trumpet music per sheet is $3

She has only $25, and she first needs to pay back $5 to her friend.

Caitlyn wanted more than $4 left over

3x < 20

3x + 4 < 20 + 4

**Step It Out**

Question 2.

Kyle is renting a boat with his family at a lake. The rental company initially charges $50 for the boat plus the hourly fee shown. The family plans to spend no more than $250 on the boat rental.

A. Describe how this problem represents an inequality situation.

Answer:

x< 8

Explanation:

50 + 25x < 250

25x < 250-50

x < 200/25

x< 8

B. What are the values in the problem that cannot change?

Answer:

The rental company initially charges $50 and boat hourly fee $25

The amount to spend no more than $250

Explanation:

From the above given information rent of a company, boat hourly fee cannot change.

C. What value in the problem can vary? How can this be represented?

Answer:

50 + 25x < 250

Explanation:

x number of hours

x is variable , time

50 + 25x < 250

D. Write an inequality for how many hours the family can rent the boat.

Answer:

x < 8 or 8 hours

Explanation:

50 + 25x < 250

25x < 250 – 50

x < 200/25

x < 8

Question 3.

A manager wants to buy headsets for the customer service team members to use. The headsets cost $12 each. The budget allows $155 to be used for customer service equipment, but the manager wants to save more than $20 for later in case it is needed for something else. How many headsets can the manager buy?

A. Will the inequality for this situation involve addition or subtraction? Explain.

Answer:

Subtraction

Explanation:

The manager budget has $155 for customer service equipment,

but the manager wants to save more than $20.

155 – 12x > 20

B. Write an inequality for how many headsets the manager can buy while staying within the limits.

Answer:

155 – 12x > 20

C. Explain why the inequality does or does not include equal to as part of the inequality symbol.

Answer:

It does not include equal to as part of the inequality symbol.

Explanation:

The inequality does or does not include equal to as part of the inequality symbol because

the manager wants to save more than $20

>$20

Question 4.

Ms. Alomry drives to school every day. Today she has 5 gallons of gas in her car. The fuel warning light comes on if the amount of gas in the tank drops below 2 gallons. She uses about \(\frac{1}{2}\) gallon of gas per round trip. How many round trips can Ms. Alomry take before her warning light turns on?

A. In this situation, is Ms. Alomry adding gas or using gas as she drives?

Answer:

using gas

B. Would the inequality have an addition or a subtraction sign in it?

Answer:

Subtraction sign in it.

C. What is the variable or the unknown in the situation?

Answer:

the number of round trips she can is a variable.

D. Ms. Alomry has to keep the gas level in her car at 2 gallons or above. Therefore, what inequality symbol should be used when referencing 2 gallons?

Answer:

Greater then or equal to

E. Write the inequality.

Answer:

5 – \(\frac{1}{2}\) t => 2

Explanation:

Ms. Alomry has 5 gallons of gas in her car.

The fuel warning light comes on if the amount of gas in the tank drops below 2 gallons.

She uses about \(\frac{1}{2}\) gallon of gas per round trip.

Total round trips can Ms. Alomry take before her warning light turns on

5 – \(\frac{1}{2}\) t => 2

**Check Understanding**

Question 1.

Gloria is saving up for the scooter shown. She has saved $63 already. She earns $7 a week for doing chores around the house. How many weeks will it take for Gloria to save at least enough money to buy the scooter? Write the inequality to represent the situation. Do not solve the inequality.

Answer:

$12.5

Explanation:

7w + 63 <= 150

7w <= 150 – 63

7w <= 87

w = 12.428

Question 2.

Write an inequality that represents 54 less than \(\frac{1}{2}\) of a number is more than 233.

Answer:

\(\frac{1}{2}\)n – 54 > 233

Explanation:

an inequality that represents 54 less than \(\frac{1}{2}\) is

\(\frac{1}{2}\)n – 54

more then 233

> 233

an inequality that represents 54 less than \(\frac{1}{2}\) of a number is more than 233 is written as

\(\frac{1}{2}\)n – 54 > 233

Question 3.

Suppose Mario either wants to earn no more than $500 or more than $500 to afford his phone. How would the inequalities be different based on these two phrases?

Answer:

m > 500

Explanation:

Mario either wants to earn no more than $500

m < 500

Mario either wants to earn no more than $500

m > 500

**On Your Own**

Question 4.

**Model with Mathematics** Saleem has $20. A fish tank that he wants to buy costs $140. He earns $23 per day at a restaurant. Saleem wants to know how many days he has to work to have at least enough money to buy the tank. What is an inequality statement that represents this situation?

Answer:

6 days

Explanation:

Saleem has $20.

A fish tank that he wants to buy costs $140.

He earns $23 per day at a restaurant.

23d + 20 = 140

23d = 140 – 20

23d = 120

d = 120/23

d = 6 days

Question 5.

**Open-Ended** Consider the inequality 3x + 7 ≥ 25. Write a word problem that represents this inequality.

Answer:

A person has 25 chocolates. He keeps 7 chocolates for himself. He plans to give 3 chocolates for each friend. How many friends can he give the chocolates to?

Explanation:

A person has 25 chocolates.

He keeps 7 chocolates for himself.

He plans to give 3 chocolates for each friend.

Number of friends can he give the chocolates to 3x + 7 ≥ 25.

Question 6.

**Model with Mathematics** There was a big snowstorm overnight that dumped 12 inches of snow on Math Town. Luckily, it got a lot warmer, and 0.5 inch is melting each hour. The snow is now less than 5 inches deep. Write an inequality to determine how many hours have passed since the snow started melting.

Answer:

12 – 0.5T < 5

Explanation:

Snow dumped = 12 inch

rate of melting of snow = 0.5 inch per hour

snow level now < 5 inch

let time taken = T

12 – T x 0.5 < 5

inequation : 12 – 0.5T < 5

**For Problems 7-10, indicate whether each value of x is a solution or is not a solution of the inequality 33x + 55 > 17.**

Question 7.

x = -2

Answer:

x is not a solution

Explanation:

33x + 55 > 17

33(-2) > 17

-66 > 17

Question 8.

x = 0

Answer:

x is a solution

Explanation:

33x + 55 > 17

33(0) + 55 > 17

0 + 55 > 17

55 > 17

Question 9.

x = -1

Answer:

x is a solution

Explanation:

33x + 55 > 17

33(-1) + 55 > 17

-33 + 55 > 17

22 > 17

Question 10.

x = 44

Answer:

x is a solution

Explanation:

33x + 55 > 17

1452 + 55 > 17

1507 > 17

**For Problems 11-14. indicate whether each value of x is a solution or is nor a solution of the inequality -9x – 4 ≤ 23.**

Question 11.

x = 3

Answer:

x is solution

Explanation:

-9x – 4 ≤ 23

x = 3

-9(3) – 4 ≤ 23

-27 – 4 ≤ 23

-31 ≤ 23

Question 12.

x = -5

Answer:

x is not a solution

Explanation:

-9x – 4 ≤ 23

-9 (- 5) – 4 ≤ 23

45 – 4 ≤ 23

41 ≤ 23

Question 13.

x = 0

Answer:

x is solution

Explanation:

-9x – 4 ≤ 23

0 – 4 ≤ 23

-4 ≤ 23

Question 14.

x = 26

Answer:

x is solution

Explanation:

-9x – 4 ≤ 23

-9 (26) – 4 ≤ 23

-238 ≤ 23

Question 15.

**Model with Mathematics** The vet says that George’s cat Milo has to lose weight, so George is going to portion out Milo’s meals and stop giving Milo table scraps. If Milo loses 0.25 pound every week, write an inequality to determine how many weeks it will take Milo to drop below 18 pounds.

Answer:

W x 0.25 < 18

Explanation :

Weight to drop below = 18 pounds

Weight lost = 0.25 per week

Let no of weeks = W

W x 0.25 < 18

W < 18 ÷ 0.25

W < 72 weeks

Question 16.

Reiko’s goal is to practice the drums for at least 800 minutes per week. This week, Reiko already has practiced for 175 minutes. If she practices for 35 minutes each session, Reiko wants to know how many sessions are left for her to make her goal.

A. Model with Mathematics Write an inequality for the situation.

Answer:

175 – S x 35 =< 800

Explanation :

Reiko’s goal is to practice the drums = 800 minutes per week.

Time Reiko already has practiced for = 175 minutes.

Time she takes for each session = 35 minutes

Let No. of sessions are left for her to make her goal = S

175 – S x 35 ≥ 800

B. Construct Arguments Explain why you chose the inequality sign that you inserted.

Answer:

in the word problem they told us that Reiko wanted to practice drums for “AT LEAST” 800 minutes, That means she can practice for 800 minutes and more so we should use ‘≥’ sign

**For Problems 17-19, write an inequality based on the statement.**

Question 17.

Twice a number plus four is at most twelve.

Answer:

2n + 4 = 12

Explanation:

Let the unknown number be n

twice a number = 2n

plus 4 is at most 12

2n + 4 = 12

Question 18.

Three less than a quarter of a number is less than six.

Answer:

\(\frac{1}{4}\)n – 3 < 6

Explanation:

Let the unknown number be n

Three less than a quarter of a number is n-3 and less than six n – 3 < 6

quarter of number \(\frac{1}{4}\)n

\(\frac{1}{4}\)n – 3 < 6

Question 19.

The sum of five and twenty-five times a number is at least one hundred.

Answer:

25n + 5 < 100

Explanation:

let the unknown number be n

The sum of five and twenty-five times a number = 25 n + 5

Question 20.

Alejandro is renting a car to visit his cousin for seven days. The rental car company charges him $199 for the week plus $0.32 per mile driven. Alejandro has $350 to spend on the rental car fees. Write an inequality to determine how many miles Alejandro can afford to drive.

Answer:

199 + M x 0.32 ≤ 350

Explanation :

Alejandro’s budget = $350

Rental charge of the car = $199

Rental charge for each mile driven = $0.32 per mile

Let No. of miles driven = M

199 + M x 0.32 ≤ 350

M x 0.32 ≤ 350 – 199

M ≤ 151 ÷ 0.32

M ≤ 471.875 miles

**I’m in a Learning Mindset!**

Do I recognize any obstacles to understanding two-step inequalities? Can those obstacles be changed? Why or why not?

Answer:

Yes solving 2 step inequality is a bit confusing. So we can solve them individually and merge them together later

**Lesson 8.2 More Practice/Homework**

Question 1.

Music Jason is a skilled rapper. He has already rapped a 40-word intro to a song and continues to rap 5 words per second. Write an inequality to determine how many more seconds t he will rap to complete the song if the total song including the intro is at least 500 words.

Answer:

500 ≤ 40 + t x 5

Explanation :

Total words = 500 at least

Words in intro = 40

Words per second = 5 words

Let time taken = t

500 ≤ 40 + t x 5

500 – 40 ≤ t x 5

460 ÷ 5 ≤ t

t >= 92 seconds

Question 2.

A team can spend no more than $300 on shirts. The team has already spent $80. How many shirts for $15 each can they still buy?

A. Write the inequality that represents the situation.

Answer:

80 + S x 15 ≤ 300

Explanation :

Team’s budget = $300

Money spent = $80

Each shirt cost = $15

Let shirts they can buy = S

80 + S x 15 ≤ 300

S x 15 ≤ 300 – 80

S ≤ 220 ÷ 15

S ≤ 14.66667

No of the shirts the team can buy with the remaining amount = 14

B. Explain why you chose the inequality sign.

Answer:

In this problem we don’t know if we are certain about our answer so if we use an inequation we can consider the best possible answer

like in the above explanation we got 14.6 as answer we can not buy 14.6 shirts thus we used ‘≤’ Since we can but shirts less than 14.6 we can buy 14 shirts which is the best possible answer

C. Without solving, explain how you know that 20 is not a solution.

Answer:

20 is not a solution as each shirt is $15 if we but 20 shirts it would cost $300

but we already used $80 which leaves us with a budget of $220

thus if we buy 20 shirts its exceeding the teams budget

Question 3.

**Model with Mathematics** Felicia spends $50 on materials to make jewelry, and she plans on selling the pieces for $7 each. A friend donates $5 to get her started. Write an inequality to find the number of pieces of jewelry Felicia can sell and make a profit.

Answer:

50 < P x 7 + 5

Explanation :

Amount Felicia spent = $50

Amount her friend gave = $5

Amount Felicia plans to sell the pieces = $7

Let no of pieces = P

50 < P x 7 + 5

Question 4.

**Math on the Spot** Write a real-world problem that could be represented by 5x + 30 ≥ 90.

Answer:

Tony buys chocolates for $90. If he sells each chocolate for $5. And his mother gave him $30 for his monthly allowance. Find how many chocolates he can sell so that there wouldn’t be any loss for him.

Question 5.

Write the inequality in words: \(\frac{1}{3}\)n – 10 > 22.

Answer:

one third of n minus ten is greater than twenty two

Question 6.

Rania is playing with her friend Arun, and they are setting up dominoes across the room. Each domino is 2 inches long. They have already placed 10 dominoes in a row. The room is 10 feet long and they want to know how many more dominoes they can set up before they reach all the way across the room. Does the inequality 2d + 20 < 120 model this situation? Explain why or why not.

Answer:

YES

Explanation :

2d + 20 < 120

length of the room = 10 feet = 10 x 12 = 120 inches

each domino length = 2 inches

dominos already placed = 10

length of dominos places = 10 x 2 = 20 inches

Let no of dominos that can be placed = d

so the equation 2d + 20 < 120 works perfectly in this situation.

**Test Prep**

Question 7.

Raphael is flying a small plane that has a weight limit of 1,200 pounds. The plane is already carrying a combined weight of 800 pounds. Each bag weighs 75 pounds. Write an inequality to find the number of bags b that can be taken on board.

Answer:

800 + b x 75 ≤ 1200

Explanation :

Weight limit = 1200 pounds

Weight already on plane = 800 pounds

Weight of each bag = 75 pounds

Let no of bags be = b

800 + b x 75 ≤ 1200

b x 75 ≤ 1200 – 800

b ≤ 400 ÷ 75

b ≤ 5.333

No of bags that the plane can carry = 5 bags

Question 8.

Select an inequality that represents 3 less than 5 times a number is no more than 63.

(A) 5n – 3 ≥ 63

(B) 5n ≥ 63 – 3

(C) 5n – 3 ≤ 63

(D) 5n ≤ 63 – 3

Answer:

Option (C)

Explanation:

3 less than 5 times a number is no more than 63

5n – 3 ≤ 63

Question 9.

Match the words with the inequality.

Answer:

Question 10.

Olivia is organizing a bake sale, and their goal is to raise at least $500. So far they have raised $210. If the items are all $5 each, which inequality represents how many more items they need to sell to meet Olivia’s goal?

(A) 5b+ 210 ≥ 500

(B) 5b ≥ 500 + 210

(C) b + 500 ≥ 210

(D) 5b+ 210 ≤ 500

Answer:

Option (D)

Explanation:

goal is to raise at least $500. So far they have raised $210.

5b+ 210 ≤ 500

**Spiral Review**

Question 11.

Max had d dollars and spent $31 but has at least $15 left. Write and solve an inequality to find d.

Answer:

d – 30 > 15

Explanation:

Max had d dollars and spent $31 but has at least $15 left

d = d – 30 > 15

Question 12.

Write a simplified expression for the perimeter of the polygon shown.

Answer:

8x + 52

Explanation:

Given;

3x + x + 7 + 18 + 2x – 4 + 30 – 2x + 4x + 1

Arrange all the variable numbers at one side and whole numbers other side

3x + x + 2x – 2x + 4x + 7 + 18 – 4 + 30 + 1

= 8x + 52