Solving Compound Inequalities (And/Or) - Step-by-Step

📚 Understanding Compound Inequalities

Compound inequalities are mathematical statements that combine two or more inequalities using the words "and" or "or". These words are crucial because they dictate how we interpret and solve the inequalities.

📜 A Brief History

The concept of inequalities dates back to ancient times, with mathematicians like Archimedes using them to approximate the value of $\pi$. The formalization of compound inequalities came later, as algebra developed, providing a concise way to express ranges and conditions.

🔑 Key Principles

• 🔍 "And" Inequalities (Intersections): These inequalities require that both conditions are true simultaneously. The solution is the overlap, or intersection, of the solutions to each individual inequality. We often write them in a compact form like $a < x < b$.
• 🤝 "Or" Inequalities (Unions): These inequalities require that at least one of the conditions is true. The solution is the combination, or union, of the solutions to each individual inequality.
• 🔢 Solving Individual Inequalities: Treat each inequality separately, using the same rules as solving regular equations, remembering to flip the inequality sign when multiplying or dividing by a negative number.
• 📈 Graphing Solutions: Represent the solution set on a number line. Use closed circles for inclusive inequalities ($\leq$ or $\geq$) and open circles for strict inequalities ($<$ or $>$).

✍️ Solving "And" Inequalities (Step-by-Step)

An "and" inequality means the solution must satisfy BOTH inequalities.

1. 🧩 Isolate the variable in each inequality separately.
2. 📈 Graph each inequality on a number line.
3. 🎨 Identify the overlapping region. This is the solution to the compound inequality.
4. ✍️ Write the solution in interval notation.

Example: Solve and graph $-3 < 2x + 1 \leq 5$

1. ➗ Isolate x (middle term):
   • Subtract 1 from all parts: $-3 - 1 < 2x + 1 - 1 \leq 5 - 1$ which simplifies to $-4 < 2x \leq 4$
   • Divide all parts by 2: $\frac{-4}{2} < \frac{2x}{2} \leq \frac{4}{2}$ which simplifies to $-2 < x \leq 2$
2. 📈 Graphing: Draw a number line. Place an open circle at -2 (since it's just '<') and a closed circle at 2 (since it's '<=').
3. 🎨 The Solution: Shade the region between -2 and 2, including 2.
4. ✍️ Interval Notation: (-2, 2]

🧪 Solving "Or" Inequalities (Step-by-Step)

An "or" inequality means the solution must satisfy AT LEAST ONE of the inequalities.

1. 🧩 Isolate the variable in each inequality separately.
2. 📈 Graph each inequality on a number line.
3. 🌈 Combine the regions. The solution includes all parts of both inequalities.
4. ✍️ Write the solution in interval notation.

Example: Solve and graph $x - 3 < -7$ or $3x + 2 \geq 5$

1. ➗ Isolate x in each inequality:
   • $x - 3 < -7$ becomes $x < -4$
   • $3x + 2 \geq 5$ becomes $3x \geq 3$, which simplifies to $x \geq 1$
2. 📈 Graphing: Draw a number line. Place an open circle at -4 (since it's '<') and a closed circle at 1 (since it's '>='). Shade to the left of -4 and to the right of 1.
3. 🌈 The Solution: The solution is everything less than -4 OR greater than or equal to 1.
4. ✍️ Interval Notation: $(-\infty, -4) \cup [1, \infty)$

🌍 Real-World Examples

• 🌡️ Temperature Ranges: A thermostat might be set to turn on the heater if the temperature is below 60°F or turn on the air conditioner if the temperature is above 80°F.
• ⚖️ Weight Limits: A bridge might have a weight limit, allowing vehicles weighing less than 5 tons and having a certain number of axles.
• 🎯 Grading Scales: A student might need to score above 90% to get an A or maintain a certain average throughout the semester.

💡 Tips for Success

• ✅ Read Carefully: Pay close attention to whether the problem uses "and" or "or". This will drastically change the solution.
• 🖍️ Graph It: Visualizing the solution on a number line can help you understand the solution set.
• 🤔 Check Your Answer: Plug values from your solution back into the original inequalities to ensure they hold true.

🔑 Conclusion

Compound inequalities provide a powerful way to express complex conditions. By understanding the key principles of "and" and "or", you can confidently solve these problems and apply them to real-world scenarios. Remember to isolate variables, graph your solutions, and always double-check your work! With practice, you'll master this important concept.