Graph Systems Of Inequalities

🔑 Key Concepts

• A system of inequalities contains two or more inequalities that are solved at the same time.
• Each inequality is graphed separately using a dashed or solid boundary line.
• A dashed line means the boundary is not included in the solution.
• A solid line means the boundary is included in the solution.
• The solution to a system of inequalities is the overlapping shaded region that satisfies all inequalities.

✏️ Worked Example — Solving and Graphing a System of Inequalities

Graphing a System of Inequalities

Problem:
Graph the following system of inequalities:

\( -x + 2y \le -4 \)
\( 3y + 6x > 18 \)

Step 1: Rewrite each inequality in slope-intercept form.

Start with:
\( -x + 2y \le -4 \)

Add \( x \) to both sides:
\( 2y \le x - 4 \)

Divide by \( 2 \):
\( y \le \frac{1}{2}x - 2 \)

Now rewrite:
\( 3y + 6x > 18 \)

Subtract \( 6x \) from both sides:
\( 3y > -6x + 18 \)

Divide by \( 3 \):
\( y > -2x + 6 \)

So the system in slope-intercept form is:
\( y \le \frac{1}{2}x - 2 \)
\( y > -2x + 6 \)

Step 2: Interpret the graph.

The line for \( y \le \frac{1}{2}x - 2 \) is drawn solid because points on that boundary are included in the solution set.

The line for \( y > -2x + 6 \) is drawn dashed because points on that boundary are not included in the solution set.

The green overlapping region is the solution to the system because every point in that region satisfies both inequalities.

A point must lie on or below the solid line and above the dashed line to be part of the solution set.

🧠 Math Vocabulary

• System of inequalities: A set of two or more inequalities solved together.
• Boundary line: The line that separates the solution region from the rest of the graph.
• Solution region: The area where all inequalities overlap.
• Dashed line: A boundary line where points on the line are not included.
• Solid line: A boundary line where points on the line are included.

💡 Main Idea

To solve a system of inequalities, graph each inequality separately and identify the region where the shaded areas overlap. That overlapping region represents all points that make every inequality in the system true at the same time.

🚀 What Comes Next

After learning to graph systems of inequalities, students will move on to solving systems algebraically, interpreting solutions in real-world contexts, and working with linear functions and constraints in modeling situations.

🧩 Embed This Video in Your LMS

Teachers can embed this video directly into Canvas, Google Classroom, Schoology, or another LMS.