Word Problem Inequalities Worksheet With Answers

Word Problem Inequalities 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 detailed explanation of how to tackle these problems, along with a variety of examples and a worksheet with answers. We'll cover different 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 review the basics of inequalities. Inequalities are mathematical statements that compare two expressions using inequality symbols:

• <: less than
• >: greater than
• ≤: less than or equal to
• ≥: greater than or equal to

Unlike equations, which have a single solution, inequalities typically have a range of solutions. For example, the inequality x > 5 means that x can be any number greater than 5.

Translating Word Problems into Inequalities

The most critical step in solving word problems involving inequalities is accurately translating the words into a mathematical expression. Look for keywords that indicate inequality:

• "At least": ≥
• "At most": ≤
• "More than": >
• "Less than": <
• "No more than": ≤
• "No less than": ≥
• "Minimum": ≥
• "Maximum": ≤
• "Exceeds": >
• "Is below": <

Let's examine some examples:

Example 1: "A number is greater than 10."

This translates to: x > 10

Example 2: "The price of a ticket is at most $25."

This translates to: p ≤ 25

Example 3: "Sarah needs at least 8 hours of sleep each night."

This translates to: s ≥ 8

Example 4: "The temperature is below freezing (0°C)."

This translates to: t < 0

Solving Inequalities

Once you've translated the word problem into an inequality, you can solve it using similar techniques to solving equations. However, remember these key differences:

• Multiplying or dividing by a negative number: When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol. For example, if * -2x < 6*, then x > -3.

• Graphing the solution: The solution to an inequality is typically a range of values. It's often helpful to graph the solution on a number line. Use an open circle (o) for inequalities with < or > and a closed circle (•) for inequalities with ≤ or ≥.

Types of Word Problems Involving Inequalities

Inequalities appear in various contexts:

1. Age Problems

Example: "John is older than his sister Mary. The sum of their ages is less than 30. If Mary is 12 years old, what are the possible ages of John?"

• Let: John's age = j, Mary's age = 12
• Inequalities: j > 12 and j + 12 < 30
• Solution: Solve the inequalities to find the range of possible ages for John.

2. Geometry Problems

Example: "The perimeter of a rectangle is at most 40 cm. The length is twice the width. Find the possible dimensions of the rectangle."

• Let: width = w, length = 2w
• Inequality: 2(w + 2w) ≤ 40
• Solution: Solve for w, then find the length.

3. Money Problems

Example: "Maria has at least $50 in her savings account. She wants to buy a new phone that costs $150. How much more money does she need to save?"

• Let: amount needed = x
• Inequality: 50 + x ≥ 150
• Solution: Solve for x.

4. Mixture Problems

Example: "A chemist needs to mix at least 10 liters of a 20% acid solution. She has two solutions available: a 10% solution and a 30% solution. How many liters of each solution should she use?"

• Let: liters of 10% solution = x, liters of 30% solution = y
• Inequalities: x + y ≥ 10 (total volume) and 0.10x + 0.30y ≥ 0.20(10) (acid concentration)
• Solution: Solve this system of inequalities. This often requires graphing the inequalities and finding the overlapping region.

Word Problem Inequalities Worksheet with Answers

Here's a worksheet with various word problems involving inequalities. Try to solve them yourself before checking the answers provided below.

Instructions: Translate each word problem into an inequality, solve it, and graph the solution on a number line.

Problem 1: The sum of a number and 5 is less than 12. Find the possible values of the number.

Problem 2: Twice a number is greater than or equal to 10. Find the possible values of the number.

Problem 3: The difference between a number and 3 is at most 7. Find the possible values of the number.

Problem 4: A rectangular garden has a length that is 3 meters more than its width. If the perimeter of the garden must be less than 26 meters, what are the possible dimensions of the garden?

Problem 5: Sarah earns $10 per hour babysitting. She needs to earn at least $150 to buy a new bike. How many hours does she need to babysit?

Problem 6: A company produces at most 500 units of a product per day. The production cost is $20 per unit, and the selling price is $35 per unit. What is the maximum profit the company can make in a day?

Problem 7: The average of two numbers is greater than 15. One number is 12. What are the possible values of the other number?

Problem 8: A triangle has sides of length x, x+2, and x+4. If the perimeter is less than 24 cm, what is the range of possible values for x?

Problem 9: A store is having a sale on books. All books are at least 20% off. If a book originally costs $25, what is the range of possible sale prices?

Problem 10: John has $100 to spend on groceries. He wants to buy at least 2 kg of apples (costing $5/kg) and at least 3 kg of oranges (costing $4/kg). How much money can he spend on other items?

Answers to the Worksheet

Problem 1: x + 5 < 12 ; x < 7

Problem 2: 2x ≥ 10 ; x ≥ 5

Problem 3: x - 3 ≤ 7 ; x ≤ 10

Problem 4: Let width = w, length = w+3. 2(w + w+3) < 26 ; 4w < 20 ; w < 5. Therefore, width < 5 meters and length < 8 meters.

Problem 5: 10h ≥ 150 ; h ≥ 15 hours.

Problem 6: Let x be the number of units produced. Profit = 35x - 20x = 15x. Since x ≤ 500, maximum profit = 15 * 500 = $7500.

Problem 7: (12 + x)/2 > 15 ; 12 + x > 30 ; x > 18

Problem 8: x + (x+2) + (x+4) < 24 ; 3x + 6 < 24 ; 3x < 18 ; x < 6

Problem 9: The discount is at least 20%, so the sale price is at most 80% of the original price. Sale price ≤ 0.80 * $25 = $20. The sale price can be anywhere between $0 and $20 inclusive.

Problem 10: Apples cost at least 2 * $5 = $10, oranges cost at least 3 * $4 = $12. Total cost of apples and oranges is at least $22. He has $100 - $22 = $78 left to spend on other items.

This comprehensive guide and worksheet provide a strong foundation for understanding and solving word problems involving inequalities. Remember to practice regularly to build your skills and confidence. By mastering these techniques, you'll be well-equipped to tackle more complex mathematical challenges.