Graphing Systems of Linear Inequalities — Understanding Overlapping Regions

🔑 Key Concepts

• A system of inequalities represents multiple conditions that must be true at the same time.
• Each inequality is graphed separately before identifying the overlapping solution region.
• An inequality with only \( x \) forms a vertical boundary line.
• A dashed (dotted) line means the boundary is not included, while a solid line means the boundary is included.
• The solution to a system of inequalities is the region where the shaded areas overlap.

✏️ Worked Example — Converting and Interpreting a System of Inequalities

Step 1: Convert Each Inequality

Let's graph the following system of inequalities:

\( x + 5 < 7 \)

\( 3y < -2x \)

---

First, isolate the variable in each inequality so it can be graphed.

\( x + 5\ {\color{red}{-5}} < 7\ {\color{red}{-5}} \)

\( x < 2 \)

---

Now convert the second inequality into graphable form.

\( \dfrac{3y}{\color{red}{3}} < \dfrac{-2x}{\color{red}{3}} \)

\( y < -\dfrac{2}{3}x \)

Step 2: Graph Each Inequality Separately

The inequality \( x < 2 \) creates a vertical boundary line because it only involves \( x \).

The inequality \( y < -\dfrac{2}{3}x \) creates a slanted boundary line.

Step 3: Identify the Overlapping Region

When both inequalities are graphed together, the overlapping region represents the solutions that satisfy both conditions.

Yellow and blue mix to form green. The green region is the solution to the system.

🧠 Math Vocabulary

• System of Inequalities: A set of two or more inequalities that must be true at the same time.
• Boundary Line: The line that separates the solutions from the non-solutions on a graph.
• Vertical Line: A line formed when an inequality contains only \( x \), such as \( x < 2 \).
• Overlapping Region: The area where the shaded regions from multiple inequalities intersect. This region represents all solutions to the system.
• Dotted Line: A boundary that is not included in the solution set because the inequality uses \( < \) or \( > \).

💡 Main Idea

To solve a system of inequalities, graph each inequality separately and then identify the region where the shaded areas overlap. That overlapping region represents all points that satisfy both inequalities at the same time. Understanding how each inequality behaves individually makes it easier to interpret the final combined graph.

📚 What You Should Already Know

Before graphing systems of inequalities, you should know how to graph a single inequality, recognize vertical and slanted lines, and interpret inequality symbols such as \( < \), \( > \), \( \le \), and \( \ge \). You should also understand how to identify slope and intercepts on a coordinate plane.

🚀 What Comes Next

After learning how to graph systems of inequalities, the next step is applying these concepts to real-world situations such as budgeting, safety zones, and resource limits. You will also explore how systems of inequalities can define feasible regions in optimization problems and decision-making scenarios.

🧩 Embed This Video in Your LMS

Teachers can embed this YouTube video directly into Canvas, Google Classroom, Schoology, or another learning platform.