How to Solve Linear Inequalities with Variables on Both Sides

📚 Understanding Linear Inequalities with Variables on Both Sides

A linear inequality is a mathematical statement that compares two expressions using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). When solving linear inequalities with variables on both sides, our goal is to isolate the variable on one side of the inequality to determine the range of values that satisfy the inequality.

📜 Historical Context

The concept of inequalities has ancient roots, appearing in early mathematical texts from various civilizations. However, the systematic study and notation of inequalities, particularly linear inequalities, developed alongside the formalization of algebra in the 16th and 17th centuries. The development of symbolic algebra allowed mathematicians to express and manipulate inequalities in a more general and abstract way, paving the way for modern applications in optimization, economics, and computer science.

🔑 Key Principles for Solving Inequalities

• ➕ Addition/Subtraction Property: ➕ You can add or subtract the same number from both sides of an inequality without changing its solution.
• ➖ Multiplication/Division by a Positive Number: ➖ You can multiply or divide both sides of an inequality by the same positive number without changing its solution.
• 🔄 Multiplication/Division by a Negative Number: 🔄 When you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality symbol. This is crucial!
• 🎯 Simplification: 🎯 Simplify each side of the inequality by combining like terms before isolating the variable.
• Isolating the Variable: Use inverse operations to isolate the variable on one side of the inequality. This usually involves adding/subtracting constants and multiplying/dividing to get the variable alone.

✍️ Step-by-Step Example

Let's solve the inequality $3x + 5 < 7x - 3$.

1. Subtract $3x$ from both sides: $5 < 4x - 3$
2. Add $3$ to both sides: $8 < 4x$
3. Divide both sides by $4$: $2 < x$
4. Rewrite: $x > 2$

Therefore, the solution to the inequality is $x > 2$.

🌍 Real-world Examples

Scenario 1: Budgeting

Suppose you want to buy some items and have a budget of $100. You already have $20 spent. Each item costs $5. How many items can you buy?

Let $x$ be the number of items. The inequality is $20 + 5x \le 100$. Solving for $x$:

• Subtract 20: $5x \le 80$
• Divide by 5: $x \le 16$

You can buy at most 16 items.

Scenario 2: Earning Money

You earn $10 per hour and need to earn at least $200. How many hours do you need to work?

Let $h$ be the number of hours. The inequality is $10h \ge 200$. Solving for $h$:

• Divide by 10: $h \ge 20$

You need to work at least 20 hours.

💡 Tips and Tricks

• 🧐 Always double-check the direction of the inequality when multiplying or dividing by a negative number.
• ✅ Verify your solution by substituting a value from your solution set back into the original inequality.
• 🎨 Use a number line to visualize the solution set.

📝 Practice Quiz

Solve the following inequalities:

1. $5x - 3 > 2x + 6$
2. $4(y + 2) \le y - 1$
3. $7 - 2z \ge 3z - 8$

Answers:

1. $x > 3$
2. $y \le -3$
3. $z \le 3$

🎯 Conclusion

Solving linear inequalities with variables on both sides is a fundamental skill in algebra. By following these principles and practicing regularly, you can master this topic and apply it to various real-world problems. Remember to pay close attention to the direction of the inequality when multiplying or dividing by a negative number. Happy solving!