Inequalities Word Problems Worksheet With Answers

Inequalities Word Problems Worksheet with Answers: A Comprehensive Guide

Solving word problems involving inequalities can be challenging, but mastering them is crucial for success in algebra and beyond. This comprehensive guide provides a structured approach to tackling these problems, complete with examples, explanations, and a worksheet with answers. We'll cover various types of inequalities, strategies for translating word problems into mathematical expressions, and techniques for solving and interpreting the solutions.

Understanding Inequalities

Before diving into word problems, let's solidify our understanding 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. The symbols used are:

• >: Greater than
• <: Less than
• ≥: Greater than or equal to
• ≤: Less than or equal to

Remember, the open end of the inequality symbol always points towards the larger value.

Translating Word Phrases into Inequalities

The key to solving inequality word problems lies in accurately translating the given information into mathematical expressions. Here's a table summarizing common word phrases and their corresponding inequality symbols:

Word Phrase	Inequality Symbol	Example
Is greater than	>	x > 5
Is less than	<	y < 10
Is greater than or equal to	≥	z ≥ 20
Is less than or equal to	≤	w ≤ 15
At least	≥	The number of apples is at least 12 (a ≥ 12)
At most	≤	The cost is at most $50 (c ≤ 50)
No more than	≤	No more than 8 students (s ≤ 8)
No less than	≥	No less than 3 hours (h ≥ 3)
Exceeds	>	The temperature exceeds 25°C (t > 25)
Is below	<	The speed is below 60 km/h (s < 60)

Strategies for Solving Inequality Word Problems

Follow these steps to effectively solve inequality word problems:

1. Read and Understand: Carefully read the problem multiple times to understand the given information and what you need to find.

2. Define Variables: Assign variables to represent the unknown quantities.

3. Translate into an Inequality: Translate the word problem into a mathematical inequality using the appropriate symbols based on the keywords.

4. Solve the Inequality: Solve the inequality using algebraic techniques, remembering to maintain the direction of the inequality sign when performing operations. (Multiplying or dividing by a negative number reverses the inequality sign).

5. Check Your Solution: Substitute your solution back into the original inequality to ensure it makes sense within the context of the problem.

6. Interpret the Solution: Write your answer in a clear and concise sentence, relating it back to the context of the word problem.

Examples of Inequality Word Problems

Let's work through some examples to solidify our understanding.

Example 1:

Maria is saving money to buy a new bicycle that costs $250. She has already saved $80 and plans to save $20 per week. How many weeks will it take for her to have enough money to buy the bicycle?

Solution:

1. Variable: Let 'w' represent the number of weeks.

2. Inequality: 80 + 20w ≥ 250

3. Solve: 20w ≥ 170 w ≥ 8.5

4. Interpret: Since Maria can't save for half a week, she needs at least 9 weeks to save enough money.

Example 2:

The temperature in a city is expected to be between 15°C and 25°C today. Express this as a compound inequality.

Solution:

Let 't' represent the temperature in °C. The inequality is 15 ≤ t ≤ 25.

Example 3:

A rectangular garden must have a perimeter of no more than 50 meters. If the length of the garden is 12 meters, what is the maximum width?

Solution:

1. Variables: Let 'w' represent the width.

2. Inequality: 2(12 + w) ≤ 50

3. Solve: 24 + 2w ≤ 50 2w ≤ 26 w ≤ 13

4. Interpret: The maximum width of the garden is 13 meters.

Example 4:

John earns $15 per hour working part-time. He needs to earn at least $300 this month to cover his expenses. How many hours must he work?

Solution:

1. Variable: Let 'h' represent the number of hours worked.

2. Inequality: 15h ≥ 300

3. Solve: h ≥ 20

4. Interpret: John must work at least 20 hours this month.

Inequalities Word Problems Worksheet

Here's a worksheet with various inequality word problems for practice. Remember to follow the steps outlined above.

(Note: The answers are provided below the worksheet.)

Problem 1: A car rental company charges $30 per day plus $0.20 per mile. If you have a budget of $100, how many miles can you drive in one day?

Problem 2: The sum of three consecutive integers is greater than 24. Find the smallest possible value for the integers.

Problem 3: Sarah is buying apples and oranges. Apples cost $0.50 each, and oranges cost $0.75 each. If she wants to spend no more than $5, and she buys 3 apples, how many oranges can she buy?

Problem 4: A triangle has two sides of length 5 cm and 7 cm. What are the possible lengths of the third side? (Hint: The triangle inequality theorem states that the sum of the lengths of any two sides must be greater than the length of the third side.)

Problem 5: A student needs to score at least an 80% average on four tests to pass the course. The student's scores on the first three tests were 75, 85, and 90. What score is needed on the fourth test to pass the course?

Answers to Worksheet

Problem 1: Let 'm' represent the number of miles. 30 + 0.20m ≤ 100; m ≤ 350 miles.

Problem 2: Let 'n' represent the smallest integer. n + (n+1) + (n+2) > 24; 3n + 3 > 24; 3n > 21; n > 7. The smallest integer is 8.

Problem 3: Let 'o' represent the number of oranges. 0.50(3) + 0.75o ≤ 5; 1.50 + 0.75o ≤ 5; 0.75o ≤ 3.50; o ≤ 4.67. Sarah can buy a maximum of 4 oranges.

Problem 4: Let 'x' represent the length of the third side. 5 + 7 > x, 5 + x > 7, 7 + x > 5. This simplifies to 2 < x < 12.

Problem 5: Let 's' represent the score on the fourth test. (75 + 85 + 90 + s)/4 ≥ 80; 250 + s ≥ 320; s ≥ 70. The student needs at least a 70 on the fourth test.

This comprehensive guide and worksheet provide a solid foundation for mastering inequality word problems. Remember to practice regularly and apply these techniques to build your confidence and problem-solving skills. By consistently practicing and understanding the underlying principles, you will become proficient in tackling these types of problems with ease. Remember to always double-check your work and ensure your answers make logical sense within the context of the word problem. Good luck!