Inequality Word Problems Worksheet With Answers Pdf

Inequality Word Problems Worksheet with Answers: A Comprehensive Guide

Solving inequality word problems can be challenging, but mastering them is crucial for success in algebra and beyond. This comprehensive guide provides a detailed walkthrough of various inequality word problem types, complete with solved examples and a downloadable worksheet (though I cannot provide actual PDF downloads, I will provide the content in a format easily printable). We'll explore strategies to effectively translate word problems into mathematical inequalities and solve them efficiently. We'll cover everything from simple inequalities to more complex scenarios involving multiple variables and constraints.

Understanding Inequalities

Before diving into word problems, let's review the basics of inequalities. Inequalities compare two expressions, indicating that one is greater than, less than, greater than or equal to, or less than or equal to the other.

• Greater than (>): x > 5 means x is any number larger than 5.
• Less than (<): y < 10 means y is any number smaller than 10.
• Greater than or equal to (≥): z ≥ 2 means z is any number larger than or equal to 2.
• Less than or equal to (≤): w ≤ -3 means w is any number smaller than or equal to -3.

Types of Inequality Word Problems

Inequality word problems appear in diverse contexts. Here are some common types:

1. Age Problems:

These problems involve comparing the ages of individuals.

Example: John is at least twice as old as his brother, Tom. If Tom is 12 years old, how old is John?

Solution:

• Let J represent John's age and T represent Tom's age.
• The problem states "John is at least twice as old as Tom," which translates to the inequality: J ≥ 2T.
• We know T = 12. Substituting this value, we get: J ≥ 2(12) => J ≥ 24.
• Therefore, John is at least 24 years old.

2. Money Problems:

These often involve budgeting, savings, or spending limits.

Example: Maria wants to buy a new phone that costs $600. She has already saved $250. If she saves $50 each week, how many weeks (w) will it take for her to have enough money to buy the phone?

Solution:

• The total amount Maria needs is $600.
• She already has $250.
• She saves $50 per week.
• The inequality representing this situation is: 250 + 50w ≥ 600
• Solving for w:
  • 50w ≥ 350
  • w ≥ 7
• It will take Maria at least 7 weeks to save enough money.

3. Distance Problems:

These often involve speed, time, and distance relationships.

Example: A car travels at a speed of at most 60 mph. If the car needs to travel 300 miles, what is the minimum time (t) it will take?

Solution:

• Distance = Speed × Time
• Distance is 300 miles, and speed is at most 60 mph.
• The inequality is: 60t ≥ 300
• Solving for t:
  • t ≥ 5
• The minimum time it will take is 5 hours.

4. Perimeter and Area Problems:

These problems involve geometric figures and their properties.

Example: The perimeter of a rectangular garden must be no more than 50 feet. If the length is 15 feet, what is the maximum width (w)?

Solution:

• Perimeter = 2(length + width)
• Perimeter ≤ 50 feet, length = 15 feet
• The inequality is: 2(15 + w) ≤ 50
• Solving for w:
  • 30 + 2w ≤ 50
  • 2w ≤ 20
  • w ≤ 10
• The maximum width is 10 feet.

5. Mixture Problems:

These problems involve combining different quantities with different properties.

Example: A chemist needs to mix a solution that is at least 20% acid. If she has 10 liters of a 10% acid solution, how many liters (x) of a 40% acid solution must she add?

Solution: This problem requires a weighted average.

• Let x be the number of liters of 40% acid solution.
• Total volume of the mixture will be 10 + x liters.
• The amount of acid in the 10% solution is 0.10 * 10 = 1 liter.
• The amount of acid in the 40% solution is 0.40 * x = 0.4x liters.
• The total amount of acid in the mixture is 1 + 0.4x liters.
• The concentration of acid in the mixture must be at least 20%, so: (1 + 0.4x) / (10 + x) ≥ 0.20
• Solving this inequality (requires cross-multiplication and careful algebraic manipulation): 1 + 0.4x ≥ 2 + 0.2x 0.2x ≥ 1 x ≥ 5
• She needs to add at least 5 liters of the 40% acid solution.

Strategies for Solving Inequality Word Problems

1. Read Carefully: Understand the problem's context and identify the unknown variable.
2. Define Variables: Assign variables to represent the unknown quantities.
3. Translate into an Inequality: Express the problem's conditions as a mathematical inequality. Pay close attention to keywords such as "at least," "at most," "more than," "less than," etc.
4. Solve the Inequality: Use algebraic techniques to isolate the variable and find the solution. Remember to reverse the inequality sign when multiplying or dividing by a negative number.
5. Check Your Answer: Verify if the solution makes sense within the context of the problem.

Inequality Word Problems Worksheet (Examples)

(Note: This is not a downloadable PDF, but the content below can easily be copied and printed.)

Instructions: Solve the following inequality word problems. Show your work.

Problem 1: Sarah is saving money to buy a bicycle that costs $200. She has already saved $80. If she saves $15 each week, how many weeks (w) will it take for her to have enough money to buy the bicycle?

Problem 2: The sum of two consecutive even integers is less than 100. What are the largest two consecutive even integers that satisfy this condition?

Problem 3: A rectangular garden has a length of 25 feet. If the perimeter must be less than 80 feet, what is the maximum width (w) of the garden?

Problem 4: A company manufactures phones. The cost to manufacture x phones is given by the function C(x) = 1000 + 20x. If the company can spend no more than $10,000 on manufacturing, how many phones can they manufacture?

Problem 5: A student needs an average score of at least 80% on three tests to pass the course. If the student scored 75% on the first test and 85% on the second test, what score (x) is needed on the third test to pass?

Problem 6: The length of a rectangle is 3 more than twice its width. If the perimeter is less than 42, what is the maximum possible width?

Problem 7: A rental car company charges $30 per day plus $0.20 per mile driven. If a customer wants to spend no more than $100, how many miles (m) can they drive in one day?

Problem 8: John has $50 to spend on snacks. Each bag of chips costs $2, and each candy bar costs $1. If he buys at least 5 bags of chips, what is the maximum number of candy bars he can buy?

Problem 9: Two angles are supplementary (they add up to 180 degrees). One angle is 20 degrees more than twice the other angle. Find the measure of the smaller angle.

Problem 10: A farmer wants to plant at least 1000 trees. He has already planted 350 apple trees and 200 pear trees. How many more orange trees (x) must he plant?

Answers (to be added after student attempts)

(This section should contain the step-by-step solutions and final answers to the above problems. This would be included in the actual downloadable worksheet.)

This comprehensive guide and worksheet will help you to master solving inequality word problems. Remember consistent practice is key. Good luck!