Solving Two Step Inequalities Word Problems

Let's unlock the secrets to solving two-step inequality word problems, transforming confusing scenarios into manageable mathematical statements, and finding the solutions you need. It's about understanding the language of math and applying it to real-world situations to reveal inequalities.

Understanding the Basics of Inequalities

Before tackling word problems, it's crucial to understand the basics of inequalities. Inequalities are mathematical statements that compare two expressions using symbols such as:

• > (greater than)
• < (less than)
• ≥ (greater than or equal to)
• ≤ (less than or equal to)

Unlike equations, which have a single solution, inequalities often have a range of solutions.

Translating Words into Mathematical Expressions

The biggest challenge in solving inequality word problems is translating the given information into a mathematical expression. Here are some keywords that often indicate inequalities:

• At least: greater than or equal to (≥)
• No less than: greater than or equal to (≥)
• More than: greater than (>)
• Exceeds: greater than (>)
• At most: less than or equal to (≤)
• No more than: less than or equal to (≤)
• Less than: less than (<)
• Below: less than (<)

Let's see how these translations work in context with examples.

Identifying the Unknown

In a word problem, it's essential to identify what you're trying to find. This unknown quantity will be represented by a variable, commonly x. Read the problem carefully to determine what x represents.

Setting Up the Inequality

Once you understand the basics of inequalities, identified the unknown variable, and understand how keywords translate to mathematical symbols, you can begin to set up the inequality. Let's go through the steps.

1. Read the problem carefully: Understand the scenario and what you're being asked to find.
2. Identify the unknown: Assign a variable (e.g., x) to represent the unknown quantity.
3. Translate keywords: Convert the words into mathematical symbols.
4. Construct the inequality: Combine the information to form a mathematical statement.

Solving Two-Step Inequalities

Once you've set up the inequality, the next step is to solve it. Solving two-step inequalities is similar to solving two-step equations, with one important difference:

• If you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.

Here are the steps to solve a two-step inequality:

1. Isolate the variable term: Add or subtract the constant term from both sides of the inequality.
2. Solve for the variable: Multiply or divide both sides of the inequality by the coefficient of the variable. Remember to reverse the inequality sign if you're multiplying or dividing by a negative number.

Checking Your Solution

After solving the inequality, it's essential to check your solution. You can do this by substituting a value from your solution set back into the original inequality. If the inequality holds true, your solution is likely correct.

Real-World Examples and Solutions

Now, let's apply these concepts to some real-world examples.

Example 1:

Problem: Sarah wants to save at least $500. She has already saved $150, and plans to save $50 each week. How many weeks will it take for her to reach her goal?

Solution:

1. Identify the unknown: Let w represent the number of weeks.
2. Set up the inequality:
   • Total savings = Initial savings + Weekly savings
   • Total savings = 150 + 50w
   • We want total savings to be at least $500, so: 150 + 50w ≥ 500
3. Solve the inequality:
   • Subtract 150 from both sides: 50w ≥ 350
   • Divide both sides by 50: w ≥ 7
4. Check the solution:
   • If w = 7, then 150 + 50(7) = 150 + 350 = 500 (Correct)
5. Answer: It will take Sarah at least 7 weeks to reach her goal.

Example 2:

Problem: A taxi charges a $3 initial fee plus $2 per mile. John wants to spend no more than $20 on his ride. How many miles can he travel?

Solution:

1. Identify the unknown: Let m represent the number of miles.
2. Set up the inequality:
   • Total cost = Initial fee + Cost per mile
   • Total cost = 3 + 2m
   • We want total cost to be no more than $20, so: 3 + 2m ≤ 20
3. Solve the inequality:
   • Subtract 3 from both sides: 2m ≤ 17
   • Divide both sides by 2: m ≤ 8.5
4. Check the solution:
   • If m = 8.5, then 3 + 2(8.5) = 3 + 17 = 20 (Correct)
5. Answer: John can travel at most 8.5 miles. Since he can't travel half a mile, he can travel 8 miles.

Example 3:

Problem: A student needs to score an average of at least 80 on three exams. She scored 75 and 85 on the first two exams. What is the minimum score she needs on the third exam?

Solution:

1. Identify the unknown: Let s represent the score on the third exam.
2. Set up the inequality:
   • Average score = (Sum of scores) / (Number of exams)
   • Average score = (75 + 85 + s) / 3
   • We want average score to be at least 80, so: (75 + 85 + s) / 3 ≥ 80
3. Solve the inequality:
   • Multiply both sides by 3: 75 + 85 + s ≥ 240
   • Simplify: 160 + s ≥ 240
   • Subtract 160 from both sides: s ≥ 80
4. Check the solution:
   • If s = 80, then (75 + 85 + 80) / 3 = 240 / 3 = 80 (Correct)
5. Answer: The student needs to score at least 80 on the third exam.

Example 4:

Problem: A rental car costs $30 per day plus $0.20 per mile. You have a budget of no more than $100. How many miles can you drive in one day?

Solution:

1. Identify the unknown: Let m represent the number of miles.
2. Set up the inequality:
   • Total cost = Daily rate + Cost per mile
   • Total cost = 30 + 0.20m
   • We want total cost to be no more than $100, so: 30 + 0.20m ≤ 100
3. Solve the inequality:
   • Subtract 30 from both sides: 0.20m ≤ 70
   • Divide both sides by 0.20: m ≤ 350
4. Check the solution:
   • If m = 350, then 30 + 0.20(350) = 30 + 70 = 100 (Correct)
5. Answer: You can drive at most 350 miles in one day.

Example 5:

Problem: A store sells notebooks for $2 each and pens for $1 each. You want to buy at least 5 notebooks and spend no more than $20. How many pens can you buy?

Solution:

1. Identify the unknown: Let p represent the number of pens. You are buying at least 5 notebooks.
2. Set up the inequality:
   • Total cost = (Cost of notebooks) + (Cost of pens)
   • Total cost = 2(5) + 1p
   • We want total cost to be no more than $20, so: 2(5) + 1p ≤ 20
3. Solve the inequality:
   • Simplify: 10 + p ≤ 20
   • Subtract 10 from both sides: p ≤ 10
4. Check the solution:
   • If p = 10, then 2(5) + 1(10) = 10 + 10 = 20 (Correct)
5. Answer: You can buy at most 10 pens.

Example 6:

Problem: You are planning a party and want to spend less than $150 on food and decorations. You spend $75 on decorations. How much can you spend on food?

Solution:

1. Identify the unknown: Let f represent the amount spent on food.
2. Set up the inequality:
   • Total cost = (Cost of decorations) + (Cost of food)
   • Total cost = 75 + f
   • We want total cost to be less than $150, so: 75 + f < 150
3. Solve the inequality:
   • Subtract 75 from both sides: f < 75
4. Check the solution:
   • If f = 74, then 75 + 74 = 149 (Correct, since 149 is less than 150)
5. Answer: You can spend less than $75 on food.

Example 7:

Problem: A car rental company charges $40 per day and $0.15 per mile. You want to rent a car for one day and spend no more than $85. How many miles can you drive?

Solution:

1. Identify the unknown: Let m represent the number of miles.
2. Set up the inequality:
   • Total cost = (Daily rate) + (Cost per mile)
   • Total cost = 40 + 0.15m
   • We want total cost to be no more than $85, so: 40 + 0.15m ≤ 85
3. Solve the inequality:
   • Subtract 40 from both sides: 0.15m ≤ 45
   • Divide both sides by 0.15: m ≤ 300
4. Check the solution:
   • If m = 300, then 40 + 0.15(300) = 40 + 45 = 85 (Correct)
5. Answer: You can drive at most 300 miles.

Example 8:

Problem: John earns $10 per hour at his job. He wants to earn at least $250 this week. How many hours must he work?

Solution:

1. Identify the unknown: Let h represent the number of hours.
2. Set up the inequality:
   • Total earnings = (Hourly rate) * (Number of hours)
   • Total earnings = 10h
   • We want total earnings to be at least $250, so: 10h ≥ 250
3. Solve the inequality:
   • Divide both sides by 10: h ≥ 25
4. Check the solution:
   • If h = 25, then 10(25) = 250 (Correct)
5. Answer: John must work at least 25 hours.

Example 9:

Problem: A school club is selling cookies to raise money. They sell each cookie for $0.75. Their goal is to raise more than $300. How many cookies must they sell?

Solution:

1. Identify the unknown: Let c represent the number of cookies.
2. Set up the inequality:
   • Total earnings = (Price per cookie) * (Number of cookies)
   • Total earnings = 0.75c
   • We want total earnings to be more than $300, so: 0.75c > 300
3. Solve the inequality:
   • Divide both sides by 0.75: c > 400
4. Check the solution:
   • If c = 401, then 0.75(401) = 300.75 (Correct, since 300.75 is more than 300)
5. Answer: They must sell more than 400 cookies.

Example 10:

Problem: A family is driving to a vacation spot 450 miles away. They have already driven 150 miles. If they average 50 miles per hour, how many more hours will it take them to reach their destination?

Solution:

1. Identify the unknown: Let h represent the number of hours.
2. Set up the inequality:
   • Remaining distance = (Total distance) - (Distance already driven)
   • Remaining distance = 450 - 150 = 300 miles
   • Distance covered in h hours = 50h
   • We want the distance covered in h hours to be at least 300 miles, so: 50h ≥ 300
3. Solve the inequality:
   • Divide both sides by 50: h ≥ 6
4. Check the solution:
   • If h = 6, then 50(6) = 300 (Correct)
5. Answer: It will take them at least 6 more hours to reach their destination.

Common Mistakes to Avoid

When solving two-step inequality word problems, there are a few common mistakes to watch out for:

• Forgetting to reverse the inequality sign: If you multiply or divide both sides of the inequality by a negative number, you must reverse the inequality sign.
• Misinterpreting keywords: Pay close attention to keywords like "at least," "no more than," "less than," and "greater than." Using the wrong symbol will lead to an incorrect solution.
• Not checking the solution: Always check your solution by substituting a value from your solution set back into the original inequality. This will help you catch any errors.
• Ignoring the context: Always consider the context of the problem when interpreting your solution. For example, if you're solving for the number of items, your answer must be a whole number.

Tips for Success

Here are some additional tips to help you succeed in solving two-step inequality word problems:

• Read the problem carefully: Make sure you understand the scenario and what you're being asked to find.
• Break the problem into smaller parts: Identify the knowns, unknowns, and the relationship between them.
• Write down each step: Show your work clearly and methodically.
• Practice regularly: The more you practice, the better you'll become at solving these types of problems.
• Seek help when needed: Don't be afraid to ask your teacher or a tutor for help if you're struggling.

Applications in Real Life

Understanding how to solve two-step inequalities is not just an academic exercise. It has many practical applications in everyday life, such as:

• Budgeting: Determining how much you can spend on different items while staying within your budget.
• Planning events: Calculating the number of guests you can invite while staying within your budget.
• Making decisions: Evaluating different options and choosing the one that best meets your needs.
• Problem-solving: Analyzing complex situations and finding solutions that meet certain criteria.

Conclusion

Solving two-step inequality word problems involves translating words into mathematical expressions, setting up and solving inequalities, and interpreting the results. By understanding the basics of inequalities, learning how to identify keywords, and practicing regularly, you can master this skill and apply it to real-world situations. Remember to read the problem carefully, show your work, check your solution, and seek help when needed. With practice and perseverance, you can conquer any inequality word problem that comes your way.