Inequality Word Problems Worksheet Answers

Tackling Inequality Word Problems: A Comprehensive Guide with Worked Examples and Answers

Understanding and solving inequality word problems is a crucial skill in algebra. These problems require translating real-world scenarios into mathematical expressions involving inequalities (>, <, ≥, ≤), and then solving those inequalities to find the solution set. This comprehensive guide will walk you through various types of inequality word problems, providing step-by-step solutions and explanations to build your confidence and mastery. We'll cover everything from basic to more complex scenarios, ensuring you're well-equipped to tackle any inequality word problem you encounter.

Understanding Inequalities: A Quick Refresher

Before diving into word problems, let's quickly review the basics of inequalities. An inequality compares two expressions, showing that one is greater than, less than, greater than or equal to, or less than or equal to the other.

• > means "greater than"
• < means "less than"
• ≥ means "greater than or equal to"
• ≤ means "less than or equal to"

Solving inequalities involves finding the range of values that satisfy the inequality. Remember that when multiplying or dividing an inequality by a negative number, you must reverse the inequality sign.

Types of Inequality Word Problems and Solution Strategies

Inequality word problems come in various forms. Here are some common types and the strategies to approach them:

1. Age Problems

Example: John is three years older than twice his sister Mary's age. If John is at least 21 years old, how old can Mary be?

Solution:

1. Define variables: Let 'x' represent Mary's age.
2. Translate into an inequality: John's age is 2x + 3. The problem states John is at least 21, so we write the inequality: 2x + 3 ≥ 21
3. Solve the inequality:
   • Subtract 3 from both sides: 2x ≥ 18
   • Divide both sides by 2: x ≥ 9
4. Interpret the solution: Mary's age (x) must be greater than or equal to 9 years old.

2. Money and Budget Problems

Example: Sarah has $50 to spend on snacks and drinks for her party. Snacks cost $10, and drinks cost $2 per bottle. If she buys 3 bags of snacks, how many bottles of drinks can she buy?

Solution:

1. Define variables: Let 'y' represent the number of drinks.
2. Translate into an inequality: The total cost is 10(3) + 2y. She has $50 to spend, so the inequality is: 10(3) + 2y ≤ 50
3. Solve the inequality:
   • Simplify: 30 + 2y ≤ 50
   • Subtract 30 from both sides: 2y ≤ 20
   • Divide both sides by 2: y ≤ 10
4. Interpret the solution: Sarah can buy at most 10 bottles of drinks.

3. Distance and Speed Problems

Example: A car travels at a speed of at least 50 mph. If the car travels for 3 hours, what is the minimum distance it covers?

Solution:

1. Define variables: Let 'd' represent the distance.
2. Translate into an inequality: Distance = Speed × Time. The inequality is: d ≥ 50 × 3
3. Solve the inequality: d ≥ 150
4. Interpret the solution: The car covers a minimum distance of 150 miles.

4. Geometry Problems

Example: The perimeter of a rectangle must be less than 20 cm. If the length is 4 cm, what are the possible values for the width?

Solution:

1. Define variables: Let 'w' represent the width.
2. Translate into an inequality: Perimeter = 2(length + width). The inequality is: 2(4 + w) < 20
3. Solve the inequality:
   • Distribute: 8 + 2w < 20
   • Subtract 8 from both sides: 2w < 12
   • Divide both sides by 2: w < 6
4. Interpret the solution: The width must be less than 6 cm.

5. Mixture Problems

Example: A chemist needs to create a solution that is at least 25% acid. If she has 10 liters of a 10% acid solution, how much pure acid (100% solution) must she add to reach the desired concentration?

Solution:

1. Define variables: Let 'x' represent the liters of pure acid added.
2. Translate into an inequality: The total amount of acid in the final solution will be 0.1(10) + x. The total volume will be 10 + x. The concentration must be at least 25%, so the inequality is: (0.1(10) + x) / (10 + x) ≥ 0.25
3. Solve the inequality:
   • Multiply both sides by (10 + x): 1 + x ≥ 2.5 + 0.25x
   • Subtract 0.25x from both sides: 0.75x ≥ 1.5
   • Divide both sides by 0.75: x ≥ 2
4. Interpret the solution: At least 2 liters of pure acid must be added.

Advanced Inequality Word Problems: Compound Inequalities

Some problems involve compound inequalities, which combine two or more inequalities. These often use "and" or "or."

Example: The temperature in a room must be between 68°F and 72°F inclusive. Write and solve the inequality representing this situation.

Solution:

This is a compound inequality using "and": 68 ≤ T ≤ 72, where T represents the temperature. The solution is any temperature between 68°F and 72°F, including 68°F and 72°F.

Troubleshooting Common Mistakes

• Incorrectly reversing the inequality sign: Remember to reverse the inequality sign only when multiplying or dividing by a negative number.
• Errors in algebraic manipulation: Carefully check each step of your algebraic manipulations to avoid errors in simplification.
• Misinterpreting the solution set: Make sure you understand what the solution set represents in the context of the word problem.

Practice Problems with Answers

Here are some practice problems to test your understanding. Try solving them on your own before checking the answers provided below.

Problem 1: A taxi charges a $3 base fare plus $2 per mile. If you have $20, how many miles can you travel?

Answer: You can travel at most 8.5 miles. (20 - 3) / 2 = 8.5

Problem 2: Maria is saving for a new bike that costs $150. She has $30 already saved and can save $15 per week. How many weeks will it take her to save enough money?

Answer: It will take her at least 8 weeks. (150 - 30) / 15 = 8

Problem 3: The average of two numbers is at least 15. If one number is 12, what is the minimum value of the other number?

Answer: The minimum value of the other number is 18. [(12 + x) / 2] ≥ 15

Problem 4: A rectangular garden must have an area of at least 50 square meters. If the length is 10 meters, what is the minimum width?

Answer: The minimum width is 5 meters. 10w ≥ 50

Problem 5: A store sells apples for $1.50 each and oranges for $1 each. You want to buy at least 5 pieces of fruit and spend no more than $6. How many apples and oranges can you buy? (Explore different combinations)

Answer: There are several possible combinations. For example, you could buy 2 apples and 3 oranges (21.5 + 31 = $6), or 1 apple and 4 oranges ($5.50), or 0 apples and 6 oranges ($6).

Conclusion

Mastering inequality word problems is a cornerstone of algebraic proficiency. By understanding the different problem types, employing the correct strategies, and practicing regularly, you can build the confidence and skills needed to solve even the most challenging problems. Remember to always clearly define variables, translate the problem into a mathematical inequality, solve carefully, and interpret your answer within the context of the original problem. Consistent practice is key to success!