Graphing and Solving a System of Inequalities

✏️ Worked Example — Converting and Interpreting a System of Inequalities

Step 1: Convert Each Inequality

Let's graph the following system of inequalities:

\( x + 5 < 7 \)

\( 3y < -2x \)

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

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

\( x < 2 \)

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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 by an inequality containing only \( x \), such as \( x < 2 \).
• Overlapping Region: The shared shaded area that satisfies every inequality in the system.
• Dashed Line: A boundary that is not included in the solution because the inequality uses \( < \) or \( > \).

📚 What You Should Already Know

Before this lesson, students should know how to solve simple inequalities, graph a single linear inequality, and read points and lines on the coordinate plane.

🚀 What Comes Next

After graphing systems of inequalities, students can move into writing inequalities from graphs, interpreting feasible regions, and comparing systems of equations to systems of inequalities.

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