8B: Word Problems from Equations & Inequalities

Which situation can be represented by the inequality

2x + 18 < 3x + 11?

• Tori is on the 18th step from the bottom of a staircase and goes up 3 steps every second. Naoki is on the 11th step from the bottom and goes up 2 steps every second. After how many seconds will Naoki pass Tori?

• Ana has made 18 bracelets and continues to make 2 bracelets each day. Elliot has made 11 bracelets and continues to make 3 bracelets each day. After how many days will Elliot have more bracelets than Ana?

• Gina has $2 in her savings account and adds $18 each week. Hiram has $3 in his savings account and adds $11 each week. After how many weeks will Gina have more money in her savings account than Hiram?

• Dhara has eaten 18 french fries and continues to eat 2 with each bite. Kendrick has eaten 11 french fries and continues to eat 3 with each bite. After how many bites will Dhara have eaten more french fries than Kendrick?

At a fruit stand, each pound of apples has a cost of x dollars. Which situation can be represented by the inequality

7x > 2x + 4?

• The cost of 7 pounds of apples is greater than the cost of 6 bunches of bananas.

• The cost of 7 pounds of apples is $4 greater than the cost of 2 pounds of apples.

• The cost of 7 pounds of apples is greater than the cost of 2 pounds of apples plus a $4 bunch of bananas.

• The cost of 7 pounds of apples is less than the cost of 2 pounds of apples plus a $4 bunch of bananas.

Which scenario could be modeled by the equation

2x + 12 = 8 + 4x?

• Jodie is wanting to join an online streaming service. She found two options online. The first option, Streaming R Us, costs $12 to activate and $2 per month for the service. The second option, Loads of Shows, costs $8 to activate the service and an additional $4 per month. At how many months will the 2 streaming options cost Jodie the same amount?

• Karl saves $2 a week of his allowance but must repay his mom $12 out of his savings for money she loaned him. Karl’s sister Tamara saves $4 a week of her allowance but must repay her mom $8 she borrowed from her out of the money she saves. After how many weeks of saving will Karl and Tamara have the same amount of money saved?

• Currently the temperature in Syracuse, NY, is −12 degrees Fahrenheit and it is 8 degrees Fahrenheit in Chicago, IL. If the temperature in Syracuse rises 4 degrees per hour and it rises 8 degrees per hour in Chicago, after how many hours will Syracuse and Chicago have the same temperature?

• Jackson swam 12 miles at a rate of 2 miles per hour and 8 miles at a rate of 4 miles per hour. How many hours does it take Jackson to swim 20 miles?

Which situation can be represented by this equation?

.5x + 10 = 2.5x + 8

• John went to an amusement park two times during the summer break. The first visit John paid $10 for the ticket with a coupon for $.50 for a drink. The second visit John paid $8 for the ticket and spent $2.50 on ice cream. What is x, the amount of time John spent at the park during the two visits?

• Jane rode her bicycle 10 miles on Saturday and stopped for a 30 minute break. The next weekend, Jane rode for 8 miles and stopped for a snow cone that cost $2.50. What is x, the time Jane spent riding her bike?

• Jane harvested 10 tomatoes from her garden on Saturday. By Tuesday, Jane was able to harvest 8 more tomatoes. Jane sold her tomatoes at the Farmer’s Market for $.50 per pound. What is x, the weight of the tomatoes in pounds?

Which situation can be represented by this inequality?

12x > 6x - 4

• The cost of 12 concert tickets is greater than the cost of 6 tickets with a $4 discount.

• The cost of 12 concert tickets with a $4 discount is greater than the cost of 6 tickets.

• The cost of 12 concert tickets is less than the cost of 6 tickets with a $4 discount.

• The cost of 12 concert tickets is greater than 4 tickets with a $6 discount.

A movie theater sells tickets for the cost of x dollars. Which situation below can be represented by this inequality?

15x < 12x + 16

• The cost of 15 tickets is greater than the cost of 12 tickets plus $16 for snacks.

• The cost of 15 tickets is less than the cost of 12 tickets plus $16 for snacks.

• The cost of 12 tickets with a coupon for $16 is less than 15 tickets.

• The cost of 15 tickets is $16 greater than the cost of 12 tickets.

Which situation can be represented by this equation?

7x + 1 = 10x

• Brody ran at a constant rate of 7 miles per hour in a race. Candice ran at a constant rate of 10 miles per hour in a race and started 1 mile in front of the starting point. What is x, the number of hours that Brody and Candice would have to run in order for their distance to be the same?

• Brody played a game where he scored 8 points per hour. Candice played the game and scored 10 points per hour. What is x, the number of hours that Brody and Candice would each have to play the game in order for their scores to be the same?

• 
  Brody went to two different amusement parks. The first park charged $7 per hour and gave a 1% discount for showing a student ID. The other park charged $10 per hour. What is x, the number of hours that Brody would have to stay at each park to have to pay the same amount?

• Brody rode his bicycle on two different days. The first day he rode 7 miles per hour and started 1 mile from his house. The second day he started from his house and rode 10 miles per hour. What is x, the number of hours that Brody would have to ride his bicycle to travel the same distance on both days?

The equation below represents the fees charged by two companies.

10 + 0.1x = 0.35x

Which scenario best describes the fees charged?

• Blue Dot Taxi charges $10 for pickup and $0.10 per mile. Yellow Check Taxi has no pickup fee but charges $0.35 per mile.

• Blue Dot Taxi charges $10 per mile and $0.10 for pickup. Yellow Check Taxi charges $0.35 for pickup.

• Blue Dot Taxi charges $10.10 for pickup. Yellow Check Taxi charges $0.35 for pickup.

• Blue Dot Taxi charges $10 per mile. Yellow Check Taxi charges $35 per mile.

The equation given below reflects the purchases made by Bill and Mary when they went to the movies.

    3x + 2.50 = 5x

Which of the following circumstances fits the relationship in the equation?

• Bill buys 3 candy bars and spends $2.50 on a soda. Mary spends the same total amount buying 5 bags of popcorn.

• Bill buys 3 candy bars for $2.50 each and Mary spends the same amount buying 5 candy bars.

• Bill buys 3 bags of popcorn and spends $2.50 on a soda. Mary spends the same amount buying 5 bags of popcorn.

• Bill pays $3 for a bag of popcorn and $2.50 for a soda. Mary spends $5 per bag on popcorn.

Which problem situation can be represented by this inequality?

25 + 12.50c ≥ 22.50c − 95

• Charles and Victor went to a comic store and purchased some comic books. Charles purchased c comic books for $12.50 each and then spent $25.00 on a comic book portfolio. Victor sold a vintage comic book to the store for $95.00 and then purchased c more comic books for $22.50 each. How many comic books, c, would each boy purchase if Charles’ total amount due is greater than or equal to Victor’s total?

• Charles and Victor went to a comic store and purchased some comic books. Charles purchased c comic books for $22.50 each and then spent $25.00 on a comic book portfolio. Victor sold a vintage comic book to the store for $95.00 and then purchased c more comic books for $12.50 each. How many comic books, c, would each boy purchase if Charles’ total amount due is greater than or equal to Victor’s total?

• Charles and Victor went to a comic store and purchased some comic books. Charles purchased c comic books for $12.50 each and then spent $25.00 on a comic book portfolio. Victor purchased a vintage comic book from the store for $95.00 and then purchased c more comic books for $22.50 each. How many comic books, c, would each boy purchase if Charles’ total amount due is greater than or equal to Victor’s total?

• Charles and Victor went to a comic store and purchased some comic books. Charles purchased c comic books for $22.50 each and then spent $25.00 on a comic book portfolio. Victor purchased a vintage comic book from the store for $95.00 and then purchased c more comic books for $12.50 each. How many comic books, c, would each boy purchase if Charles’ total amount due is greater than or equal to Victor’s total?