Algebra 1 systems of inequalities notes

📚 What are Systems of Inequalities?

A system of inequalities is a set of two or more inequalities with the same variables. The solution to a system of inequalities is the region where all the inequalities are simultaneously true. Graphically, this is the overlapping region of the graphs of each inequality.

• 📈 Inequality: A mathematical statement comparing two expressions using symbols like <, >, ≤, or ≥.
• 🧩 System: A collection of two or more inequalities considered together.
• 🎯 Solution: The set of all points that satisfy all inequalities in the system.

📜 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 application of systems of inequalities became more prominent with the development of linear programming in the 20th century, particularly in the fields of economics and operations research.

🔑 Key Principles

• ✍️ Graphing Inequalities: Each inequality is graphed on the coordinate plane. Remember to use a dashed line for strict inequalities ($<$ or $>$) and a solid line for inclusive inequalities ($≤$ or $≥$).
• 🎨 Shading: Shade the region that satisfies the inequality. For example, for $y > x + 1$, shade above the line. For $y < x + 1$, shade below the line.
• 🤝 Intersection: The solution to the system is the region where the shaded areas of all inequalities overlap. This overlapping region represents all points that satisfy all inequalities simultaneously.
• 🔎 Test Point: Choose a point in the overlapping region and substitute its coordinates into each inequality to verify that it satisfies all inequalities.

✏️ Solving Systems of Inequalities Graphically

Here’s a step-by-step guide to solving systems of inequalities graphically:

1. Graph each inequality on the same coordinate plane.
2. Identify the region where all shaded areas overlap.
3. Check your solution by picking a point in the overlapping region and verifying that it satisfies all inequalities.

➕ Example 1: A Simple System

Solve the following system of inequalities:

$\begin{cases} y > x + 1 \\ y < -x + 4 \end{cases}$

Solution:

1. Graph $y > x + 1$ (dashed line, shade above).
2. Graph $y < -x + 4$ (dashed line, shade below).
3. The solution is the region where the shaded areas overlap.

🏘️ Real-World Examples

• 💰 Budgeting: Suppose you have a budget for entertainment and dining. Let $x$ be the amount spent on entertainment and $y$ be the amount spent on dining. You might have inequalities like $x + y ≤ 100$ (total spending) and $y ≥ 2x$ (spending at least twice as much on dining as on entertainment). The solution region represents the possible spending combinations that meet your constraints.
• 🏋️ Nutrition: A nutritionist might use systems of inequalities to determine the possible combinations of foods that meet certain nutritional requirements, such as minimum amounts of vitamins and calories.
• 🏭 Manufacturing: A manufacturer might use systems of inequalities to determine the possible production levels of different products, subject to constraints on resources like labor and materials.

💡 Tips and Tricks

• ✅ Use Different Colors: When graphing, use different colors for each inequality to make the overlapping region easier to identify.
• ✏️ Label Lines: Label each line with its corresponding inequality to avoid confusion.
• 🧪 Test Points: Always test a point in the overlapping region to ensure that it satisfies all inequalities.

📝 Conclusion

Systems of inequalities are a powerful tool for modeling and solving real-world problems with constraints. By understanding the key principles and practicing with examples, you can master this important concept in Algebra 1. Keep practicing, and you'll become more confident in solving these problems!