How to Solve Two-Step Linear Inequalities: A Step-by-Step Guide

📚 What are Two-Step Linear Inequalities?

Two-step linear inequalities are mathematical expressions that involve a variable, two operations (like addition, subtraction, multiplication, or division), and an inequality sign (such as $<$, $>$, $\leq$, or $\geq$). Solving them means finding all possible values of the variable that make the inequality true.

📜 A Brief History

The concept of inequalities has been around for centuries, with early uses found in ancient Greek mathematics. However, the systematic study and notation we use today developed primarily in the 17th and 18th centuries, alongside the rise of algebra and calculus. Mathematicians like Thomas Harriot and John Wallis contributed to the symbolic representation of inequalities.

🔑 Key Principles for Solving

• ⚖️ Isolate the Variable: The main goal is to get the variable alone on one side of the inequality.
• ➕ Addition/Subtraction Principle: You can add or subtract the same number from both sides of the inequality without changing its validity.
• ➗ Multiplication/Division Principle: You can multiply or divide both sides by the same positive number. If you multiply or divide by a negative number, you must flip the inequality sign!
• 🧮 Order of Operations: Reverse the order of operations (PEMDAS/BODMAS) when isolating the variable. Undo addition/subtraction before multiplication/division.

✏️ Step-by-Step Guide

1. ➕ Step 1: Add or subtract to isolate the term with the variable.
2. ➗ Step 2: Multiply or divide to solve for the variable. Remember to flip the inequality sign if multiplying or dividing by a negative number!

💡 Example 1: Solving $2x + 3 < 7$

1. Subtract 3 from both sides: $2x + 3 - 3 < 7 - 3$ which simplifies to $2x < 4$.
2. Divide both sides by 2: $\frac{2x}{2} < \frac{4}{2}$ which simplifies to $x < 2$.
3. Therefore, the solution is all values of $x$ less than 2.

⚙️ Example 2: Solving $-3x - 5 \geq 10$

1. Add 5 to both sides: $-3x - 5 + 5 \geq 10 + 5$ which simplifies to $-3x \geq 15$.
2. Divide both sides by -3 (and flip the inequality sign!): $\frac{-3x}{-3} \leq \frac{15}{-3}$ which simplifies to $x \leq -5$.
3. Therefore, the solution is all values of $x$ less than or equal to -5.

🌍 Real-World Applications

• 🌡️ Temperature Ranges: Determining the range of temperatures within which a chemical reaction can occur.
• 💰 Budgeting: Figuring out how many items you can buy given a limited budget.
• 💪 Fitness: Calculating the range of calories you need to consume to lose or maintain weight.

✍️ Practice Quiz

1. Solve $3x - 2 > 10$.
2. Solve $-2x + 5 \leq 1$.
3. Solve $4x + 7 < 3$.
4. Solve $-5x - 3 \geq 7$.
5. Solve $\frac{x}{2} + 1 > 4$.
6. Solve $-\frac{x}{3} - 2 \leq -1$.
7. Solve $6x - 4 > 2$.

✅ Answers to Practice Quiz

1. $x > 4$
2. $x \geq 2$
3. $x < -1$
4. $x \leq -2$
5. $x > 6$
6. $x \geq -3$
7. $x > 1$

🔑 Key Takeaways

• ✔️ Solving two-step linear inequalities is similar to solving equations.
• 🔄 Remember to flip the inequality sign when multiplying or dividing by a negative number.
• 🎯 Always check your solution by plugging it back into the original inequality.

🧠 Conclusion

Mastering two-step linear inequalities is a fundamental skill in algebra. By understanding the principles and practicing regularly, you can confidently solve these problems and apply them to various real-world situations. Keep practicing, and you'll become more proficient with each problem you solve!