How To Write An Inequality From A Word Problem

Let's tackle the challenge of translating word problems into inequalities. It's a crucial skill in math, enabling us to model real-world scenarios where precise equality isn't always the case. Instead, we deal with ranges, limits, and constraints. This guide will provide you with a structured approach, key vocabulary, and examples to confidently convert word problems into mathematical inequalities.

Understanding Inequalities: The Basics

Inequalities are mathematical statements that compare two expressions using symbols other than an equal sign. The core symbols we'll be working with are:

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

The key to successfully writing inequalities from word problems lies in carefully identifying the relationships between the quantities described and choosing the appropriate symbol to represent that relationship.

Step-by-Step Guide to Writing Inequalities from Word Problems

Here's a breakdown of the process, along with examples to illustrate each step:

Step 1: Read and Understand the Problem

• Identify the Unknown: Determine what quantity you're trying to find. Assign a variable to represent it (e.g., x, y, n).
• Highlight Key Information: Underline or circle important numbers, quantities, and keywords that indicate relationships.
• Rephrase in Your Own Words: Summarize the problem in a simple sentence or two to ensure you grasp the scenario.

Example:

"A delivery driver needs to transport at least 50 boxes. Each box weighs 10 pounds. The driver's truck can carry a maximum of 1000 pounds. How many boxes can the driver transport without exceeding the truck's weight limit?"

• Unknown: The number of boxes the driver can transport. Let x = the number of boxes.
• Key Information: "at least 50 boxes," "each box weighs 10 pounds," "maximum of 1000 pounds."
• Rephrased: The driver needs to carry a minimum number of boxes, but the total weight cannot exceed the truck's capacity.

Step 2: Identify Keywords and Translate Them into Inequality Symbols

Certain words and phrases are strong indicators of inequality relationships. Here's a table of common keywords and their corresponding symbols:

Example (Continuing from Step 1):

• "at least 50 boxes" translates to x ≥ 50
• "maximum of 1000 pounds" translates to total weight ≤ 1000

Step 3: Write the Inequality

Combine the variable, the appropriate inequality symbol, and any relevant numbers or expressions to form the inequality. Be sure to consider the context of the problem and how the quantities relate to each other.

Example (Continuing from Step 2):

We know:

• x = the number of boxes
• Each box weighs 10 pounds, so the total weight is 10x
• The truck can carry a maximum of 1000 pounds.

Therefore, the inequality is: 10x ≤ 1000

And, since the driver needs to transport at least 50 boxes: x ≥ 50

So, we have a compound inequality: 50 ≤ x ≤ 100

Step 4: Check Your Inequality

• Substitute a Value: Choose a value for the variable that you believe should satisfy the inequality. Plug it into the inequality and see if the resulting statement is true.
• Consider the Context: Does the solution make sense in the real-world scenario described in the word problem? If the inequality suggests a negative number of items, that's a clear indication that something is wrong.

Example (Continuing from Step 3):

• Let's try x = 75 boxes.
• 10 * 75 = 750 pounds. 750 ≤ 1000 is true.
• 75 ≥ 50 is also true.
• 75 boxes is a reasonable solution within the context of the problem.

Step 5: Solve the Inequality (If Required)

Sometimes, the word problem asks you to find the range of possible values for the variable. In this case, you'll need to solve the inequality using algebraic techniques, remembering to reverse the inequality sign if you multiply or divide by a negative number.

Example (Continuing from Step 4):

To find the maximum number of boxes, we solve:

10x ≤ 1000

Divide both sides by 10:

x ≤ 100

Therefore, the driver can transport between 50 and 100 boxes, inclusive. 50 ≤ x ≤ 100

More Examples: Putting it All Together

Let's work through some more examples to solidify your understanding.

Example 1:

"Sarah wants to save at least $500 for a new laptop. She has already saved $150. How much more money does she need to save?"

• Unknown: The amount of money Sarah still needs to save. Let m = the amount of money.
• Key Information: "at least $500," "already saved $150."
• Rephrased: Sarah's current savings plus the additional amount she saves must be greater than or equal to $500.
• Inequality: 150 + m ≥ 500
• Check: If Sarah saves $400 more, 150 + 400 = $550, which is greater than $500.
• Solution (Solving for m): m ≥ 350. Sarah needs to save at least $350 more.

Example 2:

"A store sells bags of apples. Each bag contains 8 apples. A customer wants to buy enough bags so that they have more than 50 apples in total. How many bags of apples must they buy?"

• Unknown: The number of bags the customer must buy. Let b = the number of bags.
• Key Information: "each bag contains 8 apples," "more than 50 apples."
• Rephrased: The total number of apples (8 times the number of bags) must be greater than 50.
• Inequality: 8b > 50
• Check: If the customer buys 7 bags, 8 * 7 = 56, which is greater than 50.
• Solution (Solving for b): b > 6.25. Since you can't buy a fraction of a bag, the customer must buy at least 7 bags.

Example 3:

"A rectangle has a width of 5 cm. The area of the rectangle must be no more than 35 square cm. What is the maximum possible length of the rectangle?"

• Unknown: The length of the rectangle. Let l = the length.
• Key Information: "width of 5 cm," "area... no more than 35 square cm."
• Rephrased: The area (length times width) must be less than or equal to 35.
• Inequality: 5l ≤ 35
• Check: If the length is 6 cm, the area is 5 * 6 = 30, which is less than 35.
• Solution (Solving for l): l ≤ 7. The maximum possible length is 7 cm.

Example 4:

"John earns $12 per hour. He wants to earn at least $300 this week. How many hours must he work?"

• Unknown: The number of hours John must work. Let h = the number of hours.
• Key Information: "$12 per hour," "at least $300."
• Rephrased: John's hourly wage multiplied by the number of hours he works must be greater than or equal to $300.
• Inequality: 12h ≥ 300
• Check: If John works 26 hours, he'll earn 12 * 26 = $312, which is greater than $300.
• Solution (Solving for h): h ≥ 25. John must work at least 25 hours.

Example 5:

"The sum of a number and 7 is less than 15."

• Unknown: The number. Let n = the number.
• Key Information: "The sum of a number and 7," "is less than 15."
• Rephrased: When you add 7 to a number, the result is smaller than 15.
• Inequality: n + 7 < 15
• Check: If the number is 5, then 5 + 7 = 12, which is less than 15.
• Solution (Solving for n): n < 8. The number must be less than 8.

Advanced Scenarios and Considerations

• Compound Inequalities: Some word problems involve multiple constraints, leading to compound inequalities (like the delivery driver example). These combine two or more inequalities using "and" or "or."
• Real-World Constraints: Always consider practical limitations. For example, the number of items can't be negative, and you might need to round up or down to the nearest whole number depending on the context.
• Careful Reading: Pay close attention to the wording of the problem. A subtle change in wording can significantly alter the inequality. For instance, "less than" is different from "less than or equal to."
• Multi-Step Problems: Some problems require you to perform calculations before you can write the inequality. Break down the problem into smaller steps.

Common Mistakes to Avoid

• Incorrect Symbol: Choosing the wrong inequality symbol is a common error. Double-check the keywords to ensure you're using the correct symbol.
• Misinterpreting the Relationship: Carefully analyze how the quantities are related. Are you dealing with a maximum, minimum, or a range of values?
• Ignoring Units: Make sure your units are consistent. If you're dealing with different units, convert them to the same unit before writing the inequality.
• Forgetting Real-World Constraints: Always consider the practicality of the solution within the context of the problem.

Practice Problems

Here are some practice problems to test your skills. Try to solve them on your own, and then check your answers:

1. A student needs an average score of at least 80 on four tests to get a B. The student scored 75, 82, and 85 on the first three tests. What is the minimum score the student needs on the fourth test?
2. A phone company charges $0.10 per minute for calls, plus a monthly fee of $15. A customer wants to keep their monthly bill no more than $25. What is the maximum number of minutes they can use?
3. The perimeter of a square must be greater than 40 inches. What is the minimum length of one side of the square?
4. A taxi charges a flat fee of $3, plus $2 per mile. A passenger wants to spend no more than $15 on a ride. What is the maximum number of miles they can travel?
5. The difference between twice a number and 5 is greater than 11. What are the possible values of the number?

Solutions to Practice Problems

1. Let x be the score on the fourth test. The inequality is (75 + 82 + 85 + x) / 4 ≥ 80. Solving for x, we get x ≥ 78. The student needs to score at least 78 on the fourth test.
2. Let m be the number of minutes. The inequality is 15 + 0.10m ≤ 25. Solving for m, we get m ≤ 100. The customer can use a maximum of 100 minutes.
3. Let s be the length of one side of the square. The perimeter is 4s. The inequality is 4s > 40. Solving for s, we get s > 10. The minimum length of one side is greater than 10 inches.
4. Let m be the number of miles. The inequality is 3 + 2m ≤ 15. Solving for m, we get m ≤ 6. The passenger can travel a maximum of 6 miles.
5. Let n be the number. The inequality is 2n - 5 > 11. Solving for n, we get n > 8. The number must be greater than 8.

Conclusion

Writing inequalities from word problems is a fundamental skill that connects mathematics to real-world situations. By understanding the basic concepts, recognizing keywords, and following a structured approach, you can confidently translate word problems into mathematical inequalities and solve them effectively. Remember to practice regularly and always consider the context of the problem to ensure your solutions are meaningful. Mastering this skill will empower you to tackle a wide range of mathematical challenges.