Inequality Word Problems Worksheet And Answers

Inequality Word Problems Worksheet and Answers: A Comprehensive Guide

Inequalities are a crucial part of algebra, and understanding how to solve them is vital for success in mathematics and related fields. Word problems, however, often present a challenge. This comprehensive guide provides a detailed explanation of how to tackle inequality word problems, along with numerous examples and solutions to help solidify your understanding. We'll cover various types of inequalities and strategies to solve them effectively. By the end, you'll be equipped to confidently approach any inequality word problem.

Understanding Inequalities

Before diving into word problems, let's review the basics of inequalities. An inequality shows the relationship between two expressions that are not equal. The symbols used are:

• > greater than
• < less than
• ≥ greater than or equal to
• ≤ less than or equal to
• ≠ not equal to

Unlike equations, inequalities have multiple solutions, represented by a range of values.

Types of Inequality Word Problems

Inequality word problems appear in various forms, each requiring a slightly different approach. Let's categorize them:

1. Comparison Problems:

These problems involve comparing two quantities using inequality symbols.

Example: John's age is more than twice Mary's age. If Mary is 10 years old, what are the possible ages of John?

Solution:

Let J represent John's age and M represent Mary's age. The problem states: J > 2M. Since M = 10, the inequality becomes: J > 2(10), which simplifies to J > 20. Therefore, John's age is greater than 20 years old.

2. Minimum/Maximum Value Problems:

These problems involve finding the smallest or largest possible value that satisfies certain conditions.

Example: A company needs to manufacture at least 500 units of a product each day. They can produce x units using Machine A and y units using Machine B. If Machine A produces 200 units and Machine B produces 150 units, write an inequality representing the situation.

Solution:

The total units produced must be at least 500. Therefore, the inequality is: x + y ≥ 500. In this case, 200 + 150 = 350 which is less than 500, meaning more units are needed.

3. Range Problems:

These problems involve determining a range of values that satisfy given conditions.

Example: The temperature in a city is expected to be between 15°C and 25°C today. Write an inequality to represent this situation.

Solution:

Let T represent the temperature. The inequality is: 15 ≤ T ≤ 25.

4. Rate and Distance Problems:

These problems often involve speed, time, and distance, using inequalities to represent constraints.

Example: A car travels at a speed of at least 60 mph. If the distance is 300 miles, what is the maximum time it can take?

Solution:

Speed = Distance / Time. We have speed ≥ 60 mph and distance = 300 miles. Let T represent time in hours. The inequality becomes: 60 ≤ 300/T. Solving for T, we get T ≤ 5 hours.

5. Budget/Cost Problems:

These problems involve managing expenses within a budget, using inequalities to represent constraints.

Example: Sarah has a budget of $100 to buy apples and oranges. Apples cost $2 each, and oranges cost $3 each. Let x represent the number of apples and y represent the number of oranges. Write an inequality to represent her spending limit.

Solution:

The total cost of apples and oranges must be less than or equal to $100. The inequality is: 2x + 3y ≤ 100.

Solving Inequality Word Problems: A Step-by-Step Guide

1. Read and Understand: Carefully read the problem several times to understand the situation and the relationships between the variables. Identify keywords such as "at least," "at most," "more than," "less than," "between," etc.

2. Define Variables: Assign variables to the unknown quantities. Clearly state what each variable represents.

3. Write the Inequality: Translate the problem's information into a mathematical inequality using the appropriate symbols.

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. Check Your Solution: Substitute your solution back into the original inequality to ensure it satisfies the conditions of the problem. Consider the context: negative values might not make sense in real-world situations.

6. State Your Answer: Clearly state your answer in the context of the original problem.

Examples with Detailed Solutions

Let's work through several examples illustrating different types of inequality word problems.

Example 1: The Sum of Two Numbers

Two numbers, x and y, have a sum that is greater than 10. If x is 3, what are the possible values of y?

Solution:

1. Read and Understand: We need to find the possible values of y given that x + y > 10 and x = 3.

2. Define Variables: x = 3, y = unknown number.

3. Write the Inequality: 3 + y > 10

4. Solve the Inequality: Subtract 3 from both sides: y > 7.

5. Check: If y = 8, then 3 + 8 = 11 > 10 (True).

6. State Your Answer: The possible values of y are greater than 7.

Example 2: A Rental Car

A rental car company charges a flat fee of $30 plus $0.25 per mile driven. If you have a budget of $100, how many miles can you drive?

Solution:

1. Read and Understand: We need to find the maximum number of miles that can be driven within a $100 budget.

2. Define Variables: Let m be the number of miles driven.

3. Write the Inequality: 30 + 0.25m ≤ 100

4. Solve the Inequality: Subtract 30 from both sides: 0.25m ≤ 70 Divide both sides by 0.25: m ≤ 280

5. Check: If m = 280, then 30 + 0.25(280) = 100 (True).

6. State Your Answer: You can drive a maximum of 280 miles.

Example 3: Exam Scores

To pass a course, a student needs an average score of at least 70% on three exams. If the student scored 65% and 78% on the first two exams, what is the minimum score needed on the third exam to pass?

Solution:

1. Read and Understand: We need to find the minimum score on the third exam to achieve an average of at least 70%.

2. Define Variables: Let x be the score on the third exam.

3. Write the Inequality: (65 + 78 + x)/3 ≥ 70

4. Solve the Inequality: (143 + x)/3 ≥ 70 143 + x ≥ 210 x ≥ 67

5. Check: If x = 67, then (65 + 78 + 67)/3 = 70 (True).

6. State Your Answer: The student needs a minimum score of 67% on the third exam.

Conclusion

Solving inequality word problems requires a systematic approach. By carefully reading, defining variables, writing the appropriate inequality, solving it correctly, checking your answer, and stating it clearly, you can master this important algebraic skill. Practice is key! The more examples you work through, the more confident you will become in tackling any inequality word problem you encounter. Remember to always consider the context of the problem to ensure your solution is realistic and meaningful.