Graph to Solve Systems of Inequalities

🔑 Key Concepts

• A system of inequalities is a set of two or more inequalities graphed on the same coordinate plane.
• Each inequality has its own boundary line and shaded half-plane.
• A solid boundary line means points on the line are included in the solution set.
• A dashed boundary line means points on the line are not included in the solution set.
• The final solution is the overlapping shaded region, where points satisfy every inequality in the system.

✏️ 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 blue-only region represents points that satisfy \( y \le \frac{1}{2}x - 2 \) but do not satisfy the other inequality.

The yellow-only region represents points that satisfy \( y > -2x + 6 \) but do not satisfy the first inequality.

The green overlapping region is the most important part of the graph. Every point in that region satisfies both inequalities at the same time, so that overlap is the solution to the system.

In other words, 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: two or more inequalities considered together on the same graph.
• Boundary Line: the line that separates solutions from non-solutions for an inequality.
• Solid Line: shows that points on the boundary are included.
• Dashed Line: shows that points on the boundary are not included.
• Overlap: the shared region where all inequalities in the system are true.

💡 Main Idea

To graph a system of inequalities, first rewrite each inequality in slope-intercept form when needed. Then graph each boundary line and shade the region that satisfies each individual inequality. A solid line is used when the boundary is included, while a dashed line is used when it is not. The final solution is the overlapping shaded region, because every point in that intersection satisfies the entire system.

📚 What You Should Already Know

Students should already know how to solve linear equations, rewrite equations in slope-intercept form, graph lines on the coordinate plane, and distinguish between strict and inclusive inequality symbols.

🚀 What Comes Next

After graphing systems of inequalities, students can analyze solution regions, test ordered pairs, model real-world constraints, and connect systems of inequalities to optimization and linear programming ideas in more advanced mathematics.

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