steps to solve linear inequalities

📚 Understanding Linear Inequalities

A linear inequality is a mathematical statement that compares two expressions using inequality symbols. Unlike equations that use an equals sign (=), inequalities use symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Solving linear inequalities involves finding the range of values that satisfy the inequality.

📜 History and Background

The concept of inequalities has been around for centuries, with early uses found in ancient Greek mathematics. However, the systematic study and notation of inequalities, as we know it today, developed more fully in the 17th and 18th centuries. Mathematicians like Thomas Harriot and John Wallis contributed to the symbolic representation and understanding of inequalities.

🔑 Key Principles for Solving Linear Inequalities

• ⚖️ Addition and Subtraction: You can add or subtract the same number from both sides of an inequality without changing the solution.
• ✖️ Multiplication and Division by a Positive Number: You can multiply or divide both sides of an inequality by the same positive number without changing the solution.
• 🔄 Multiplication and 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 sign. This is a crucial step!
• 📈 Simplification: Simplify each side of the inequality by combining like terms before applying any operations.
• 📊 Graphing: Represent the solution set on a number line to visualize the range of values that satisfy the inequality.

🪜 Steps to Solve Linear Inequalities

1. Simplify Both Sides: Combine like terms on each side of the inequality.
2. Isolate the Variable Term: Use addition or subtraction to get the variable term on one side and the constant terms on the other.
3. Solve for the Variable: Use multiplication or division to isolate the variable. Remember to flip the inequality sign if you multiply or divide by a negative number.
4. Check Your Solution: Substitute a value from your solution set back into the original inequality to make sure it holds true.
5. Graph the Solution: Represent the solution on a number line. Use an open circle for < and >, and a closed circle for ≤ and ≥.

✍️ Example 1: Solving a Simple Linear Inequality

Solve the inequality: $3x + 5 < 14$

1. Subtract 5 from both sides: $3x < 9$
2. Divide both sides by 3: $x < 3$

Solution: $x < 3$. This means any value of x less than 3 will satisfy the inequality.

➗ Example 2: Solving with a Negative Coefficient

Solve the inequality: $-2x + 7 ≥ 1$

1. Subtract 7 from both sides: $-2x ≥ -6$
2. Divide both sides by -2 (and flip the inequality sign): $x ≤ 3$

Solution: $x ≤ 3$. Remember to flip the inequality sign when dividing by a negative number!

➕ Example 3: Solving with Distribution

Solve the inequality: $2(x - 1) > 3x + 4$

1. Distribute the 2: $2x - 2 > 3x + 4$
2. Subtract 2x from both sides: $-2 > x + 4$
3. Subtract 4 from both sides: $-6 > x$
4. Rewrite: $x < -6$

Solution: $x < -6$

💡 Tips and Tricks

• ✔️ Double-Check: Always double-check your work, especially when multiplying or dividing by a negative number.
• 🧭 Number Line: Use a number line to visualize the solution set. This can help prevent errors.
• 🧮 Substitution: Substitute a value from your solution set into the original inequality to verify your answer.

🌍 Real-World Applications

• 🌡️ Temperature Ranges: Determining the range of temperatures within which a certain chemical reaction will occur.
• 💰 Budgeting: Calculating how much money you can spend each week while staying within your budget.
• 🏋️ Fitness: Setting target heart rate zones during exercise.

📝 Conclusion

Solving linear inequalities is a fundamental skill in algebra with many practical applications. By following these steps and understanding the key principles, you can confidently solve a wide range of inequalities. Remember to pay close attention to the direction of the inequality sign, especially when multiplying or dividing by negative numbers. Keep practicing, and you'll master this skill in no time!