How To Do Inequality Word Problems

Inequality word problems can be tricky, but they're also a powerful tool for understanding real-world constraints and possibilities. They help us move beyond simple equations to situations where a range of solutions is acceptable. This guide will walk you through the process of solving inequality word problems, providing clear steps, examples, and practical tips to master this skill.

Understanding the Basics of Inequalities

Before diving into word problems, it's crucial to understand the fundamental symbols and concepts of inequalities:

• > (Greater than): One value is larger than another. For example, "x > 5" means x is greater than 5.
• < (Less than): One value is smaller than another. For example, "y < 10" means y is less than 10.
• ≥ (Greater than or equal to): One value is larger than or equal to another. For example, "z ≥ 3" means z is greater than or equal to 3.
• ≤ (Less than or equal to): One value is smaller than or equal to another. For example, "w ≤ 7" means w is less than or equal to 7.

These symbols represent relationships where values are not necessarily equal but exist within a certain range. This range is what makes inequalities so useful for modeling real-world scenarios with limitations or minimum requirements.

Steps to Solve Inequality Word Problems

Solving inequality word problems requires a systematic approach. Here's a breakdown of the key steps:

1. Read and Understand the Problem: Carefully read the problem statement. Identify what information is provided and what you are asked to find. Look for keywords that indicate inequality relationships.
2. Define Variables: Assign variables to represent the unknown quantities in the problem. Clearly state what each variable represents.
3. Translate into an Inequality: Convert the word problem into a mathematical inequality. Use keywords and relationships described in the problem to write the inequality.
4. Solve the Inequality: Use algebraic techniques to solve the inequality for the variable. Remember to reverse the inequality sign if you multiply or divide by a negative number.
5. Interpret the Solution: Translate the solution back into the context of the word problem. State the meaning of the solution in terms of the original question.
6. Check Your Solution: Verify that your solution makes sense in the context of the problem. Substitute values within the solution range back into the original problem to ensure they satisfy the given conditions.

Keywords for Inequalities

Certain keywords often appear in inequality word problems, signaling the type of inequality to use. Here's a table summarizing common keywords:

Example Problems and Solutions

Let's work through several example problems to illustrate the steps involved in solving inequality word problems:

Example 1: The Concert Ticket Problem

Problem: You want to buy concert tickets that cost $25 each. You have saved $150. What is the maximum number of tickets you can buy?

1. Understand the Problem: We need to find the maximum number of tickets you can buy with $150.
2. Define Variables: Let t represent the number of tickets.
3. Translate into an Inequality: The cost of t tickets at $25 each must be less than or equal to the amount you have saved. So, 25t ≤ 150.
4. Solve the Inequality: Divide both sides of the inequality by 25:
   • 25t/25 ≤ 150/25
   • t ≤ 6
5. Interpret the Solution: You can buy at most 6 tickets.
6. Check Your Solution:
   • If you buy 6 tickets, the cost is 6 * $25 = $150.
   • If you buy 7 tickets, the cost is 7 * $25 = $175, which is more than you have.

Example 2: The Restaurant Budget Problem

Problem: A restaurant needs to spend less than $500 on supplies each week. They buy flour at $5 per bag and sugar at $8 per bag. If they buy 40 bags of flour, how many bags of sugar can they buy?

1. Understand the Problem: We need to find the maximum number of bags of sugar the restaurant can buy given their budget and flour purchase.
2. Define Variables: Let s represent the number of bags of sugar.
3. Translate into an Inequality: The cost of flour plus the cost of sugar must be less than $500. So, (40 * 5) + 8s < 500.
4. Solve the Inequality:
   • 200 + 8s < 500
   • 8s < 500 - 200
   • 8s < 300
   • s < 300/8
   • s < 37.5
5. Interpret the Solution: The restaurant can buy less than 37.5 bags of sugar. Since they can't buy half a bag, they can buy at most 37 bags of sugar.
6. Check Your Solution:
   • If they buy 37 bags of sugar, the cost is (40 * $5) + (37 * $8) = $200 + $296 = $496, which is less than $500.
   • If they buy 38 bags of sugar, the cost is (40 * $5) + (38 * $8) = $200 + $304 = $504, which is more than $500.

Example 3: The Test Score Problem

Problem: To get an A in a course, a student needs an average of at least 90 on four tests. The student scored 85, 92, and 88 on the first three tests. What is the minimum score the student needs on the fourth test to get an A?

1. Understand the Problem: We need to find the minimum score needed on the fourth test to achieve an average of at least 90.
2. Define Variables: Let x represent the score on the fourth test.
3. Translate into an Inequality: The average of the four test scores must be greater than or equal to 90. So, (85 + 92 + 88 + x)/4 ≥ 90.
4. Solve the Inequality:
   • (265 + x)/4 ≥ 90
   • 265 + x ≥ 360
   • x ≥ 360 - 265
   • x ≥ 95
5. Interpret the Solution: The student needs to score at least 95 on the fourth test to get an A.
6. Check Your Solution:
   • If the student scores 95, the average is (85 + 92 + 88 + 95)/4 = 360/4 = 90.
   • If the student scores 94, the average is (85 + 92 + 88 + 94)/4 = 359/4 = 89.75, which is less than 90.

Example 4: The Car Rental Problem

Problem: A car rental company charges $30 per day plus $0.20 per mile. You want to rent a car for a day and stay within a budget of $80. What is the maximum number of miles you can drive?

1. Understand the Problem: We need to find the maximum number of miles you can drive while staying within a budget of $80.
2. Define Variables: Let m represent the number of miles driven.
3. Translate into an Inequality: The daily charge plus the mileage charge must be less than or equal to $80. So, 30 + 0.20m ≤ 80.
4. Solve the Inequality:
   • 0. 20m ≤ 80 - 30
   • 0. 20m ≤ 50
   • m ≤ 50/0.20
   • m ≤ 250
5. Interpret the Solution: You can drive at most 250 miles.
6. Check Your Solution:
   • If you drive 250 miles, the cost is $30 + (250 * $0.20) = $30 + $50 = $80.
   • If you drive 251 miles, the cost is $30 + (251 * $0.20) = $30 + $50.20 = $80.20, which is more than $80.

Example 5: The Fundraiser Problem

Problem: A club is selling cookies to raise money. They earn $2 for each small box and $5 for each large box. They want to raise at least $300. If they sell 50 small boxes, how many large boxes do they need to sell?

1. Understand the Problem: We need to find the minimum number of large boxes they need to sell to raise at least $300, given that they sell 50 small boxes.
2. Define Variables: Let l represent the number of large boxes.
3. Translate into an Inequality: The earnings from small boxes plus the earnings from large boxes must be greater than or equal to $300. So, (50 * 2) + 5l ≥ 300.
4. Solve the Inequality:
   • 100 + 5l ≥ 300
   • 5l ≥ 300 - 100
   • 5l ≥ 200
   • l ≥ 200/5
   • l ≥ 40
5. Interpret the Solution: They need to sell at least 40 large boxes.
6. Check Your Solution:
   • If they sell 40 large boxes, the earnings are (50 * $2) + (40 * $5) = $100 + $200 = $300.
   • If they sell 39 large boxes, the earnings are (50 * $2) + (39 * $5) = $100 + $195 = $295, which is less than $300.

Tips for Solving Inequality Word Problems

• Read Carefully and Multiple Times: Understand the problem thoroughly before attempting to solve it.
• Highlight Key Information: Identify and highlight important numbers, relationships, and keywords.
• Draw Diagrams or Charts: Visual aids can help you organize information and understand relationships.
• Break Down Complex Problems: Divide complex problems into smaller, more manageable parts.
• Practice Regularly: The more you practice, the better you will become at recognizing patterns and applying the correct strategies.
• Pay Attention to Units: Ensure that all units are consistent throughout the problem.
• Check Your Answer: Always verify that your solution makes sense in the context of the original problem.
• Consider Real-World Constraints: Remember that solutions must be practical. For example, you can't buy a fraction of a ticket or sell a negative number of boxes.

Common Mistakes to Avoid

• Misinterpreting Keywords: Incorrectly identifying keywords can lead to the wrong inequality.
• Forgetting to Reverse the Inequality Sign: Remember to reverse the inequality sign when multiplying or dividing by a negative number.
• Incorrectly Defining Variables: Clearly define what each variable represents to avoid confusion.
• Ignoring Real-World Constraints: Failing to consider practical limitations can lead to unrealistic solutions.
• Not Checking the Solution: Always verify that your solution makes sense in the context of the problem.
• Algebraic Errors: Double-check your algebraic manipulations to avoid errors.

Advanced Inequality Problems

Once you're comfortable with basic inequality word problems, you can tackle more complex scenarios involving multiple variables, systems of inequalities, and compound inequalities. These types of problems require a deeper understanding of algebraic techniques and problem-solving strategies.

Example: The Investment Problem

Problem: An investor wants to invest up to $20,000 in two different accounts. Account A yields 5% interest per year, and Account B yields 8% interest per year. The investor wants to earn at least $1200 in interest per year. What is the maximum amount the investor can invest in Account A?

1. Understand the Problem: We need to find the maximum amount the investor can invest in Account A while staying within a total investment limit and earning a minimum amount of interest.
2. Define Variables: Let a represent the amount invested in Account A, and b represent the amount invested in Account B.
3. Translate into Inequalities:
   • a + b ≤ 20000 (Total investment limit)
   • 0.05a + 0.08b ≥ 1200 (Minimum interest earned)
4. Solve the Inequalities:
   • From the first inequality, we can express b in terms of a: b ≤ 20000 - a
   • Substitute this expression for b into the second inequality:
     • 0.05a + 0.08(20000 - a) ≥ 1200
     • 0.05a + 1600 - 0.08a ≥ 1200
     • -0.03a ≥ -400
     • a ≤ 400/0.03 (Remember to reverse the inequality sign when dividing by a negative number)
     • a ≤ 13333.33
5. Interpret the Solution: The investor can invest at most $13,333.33 in Account A.
6. Check Your Solution:
   • If the investor invests $13,333.33 in Account A, then they invest $6,666.67 in Account B.
   • The interest earned is (0.05 * $13,333.33) + (0.08 * $6,666.67) = $666.67 + $533.33 = $1200.

Conclusion

Solving inequality word problems is a valuable skill that can be applied in various real-world scenarios. By understanding the basics of inequalities, following a systematic approach, and practicing regularly, you can master this skill and gain confidence in your problem-solving abilities. Remember to read carefully, define variables clearly, translate accurately, and always check your solution to ensure it makes sense. With persistence and practice, you'll be able to tackle even the most challenging inequality word problems.