Solved Problems: Translating and Solving Linear Inequality Word Problems

📚 Understanding Linear Inequality Word Problems

Linear inequality word problems involve translating real-world scenarios into mathematical statements that use inequality symbols ($<, >, \leq, \geq$). Solving these problems requires careful reading, identifying key information, and applying algebraic techniques.

📜 A Brief History

The use of inequalities in mathematics dates back centuries, with early applications in geometry and number theory. However, the systematic study and application of linear inequalities to solve practical problems gained prominence with the development of linear programming in the 20th century. Linear programming techniques are widely used in economics, engineering, and operations research to optimize resource allocation and decision-making.

🔑 Key Principles for Translation

• 🔍 Identify Variables: Define what the variables represent in the problem. For example, let $x$ be the number of hours worked.
• ✍️ Translate Key Phrases: Recognize phrases that indicate inequality. Common phrases include:
• "More than" translates to $>$.
• "Less than" translates to $<$.
• "At least" translates to $\geq$.
• "At most" translates to $\leq$.
• "No more than" translates to $\leq$.
• "Exceeds" translates to $>$.
• ➕ Formulate the Inequality: Combine the variables and constants using the appropriate inequality symbol.
• 💡 Solve the Inequality: Use algebraic techniques to isolate the variable and find the solution set.
• ✅ Check Your Solution: Verify that the solution satisfies the original word problem.

🌍 Real-world Examples

Example 1: Budgeting

A student wants to save at least $500 for a trip. They have already saved $200 and plan to save $30 per week. How many weeks will it take?

1. Variable: Let $w$ be the number of weeks.
2. Inequality: $200 + 30w \geq 500$
3. Solution: $30w \geq 300$ $w \geq 10$
4. Answer: It will take at least 10 weeks.

Example 2: Concert Tickets

A family wants to spend no more than $400 on concert tickets. Tickets cost $50 for adults and $30 for children. If there are 2 adults, how many children can attend?

1. Variable: Let $c$ be the number of children.
2. Inequality: $(2 \times 50) + 30c \leq 400$
3. Solution: $100 + 30c \leq 400$ $30c \leq 300$ $c \leq 10$
4. Answer: At most 10 children can attend.

Example 3: Earning Money

A person earns $15 per hour. How many hours must they work to earn more than $300?

1. Variable: Let $h$ be the number of hours.
2. Inequality: $15h > 300$
3. Solution: $h > \frac{300}{15}$ $h > 20$
4. Answer: They must work more than 20 hours.

📝 Practice Quiz

1. John wants to buy a car that costs no more than $25,000. He has $5,000 saved and can save $500 each month. How many months will it take?

   Solution: Let $m$ be the number of months. $5000 + 500m \leq 25000$. Solving for $m$ gives $m \leq 40$. It will take at most 40 months.

2. A store sells apples for $2 each and bananas for $1 each. A customer wants to spend at least $10. If they buy 3 apples, how many bananas must they buy?

   Solution: Let $b$ be the number of bananas. $(3 \times 2) + 1b \geq 10$. Solving for $b$ gives $b \geq 4$. They must buy at least 4 bananas.

3. A person wants to run more than 10 miles each week. They have already run 3 miles. How many more miles must they run each week?

   Solution: Let $m$ be the additional miles. $3 + m > 10$. Solving for $m$ gives $m > 7$. They must run more than 7 miles.

4. A teacher wants to spend at most $50 on supplies. Each pen costs $2, and each notebook costs $3. If the teacher buys 10 pens, how many notebooks can they buy?

   Solution: Let $n$ be the number of notebooks. $(10 \times 2) + 3n \leq 50$. Solving for $n$ gives $n \leq 10$. They can buy at most 10 notebooks.

5. A student needs to score at least 80 points on the final exam to pass the course. The exam is worth 100 points. If the student has already earned 60 points, what is the minimum score they need on the final exam?

   Solution: Let $s$ be the score on the final exam. $60 + s \geq 80$. Solving for $s$ gives $s \geq 20$. They need to score at least 20 points.

6. A company wants to produce at least 500 units of a product. They can produce 50 units per day. How many days will it take?

   Solution: Let $d$ be the number of days. $50d \geq 500$. Solving for $d$ gives $d \geq 10$. It will take at least 10 days.

7. A person wants to save more than $1000. They have $200 saved and can save $50 each week. How many weeks will it take?

   Solution: Let $w$ be the number of weeks. $200 + 50w > 1000$. Solving for $w$ gives $w > 16$. It will take more than 16 weeks.

🎓 Conclusion

Translating and solving linear inequality word problems is a valuable skill in mathematics with applications in various real-world scenarios. By understanding the key principles and practicing with examples, you can master this topic and confidently tackle challenging problems.