Graph System of Inequalities- Step-by-Step Tutorial

What Is a System of Inequalities?

A system of inequalities is just two or more inequalities graphed on the same coordinate plane. The solution isn't a single point—it's the region where all inequalities overlap.

That's it. Nothing fancy. You graph each inequality separately, find where they all agree, and that overlapping zone is your answer.

Before You Start: The Basics You Need to Know

If you can't graph a single inequality, you're going to struggle with systems. Here's what should already be in your toolkit:

• How to graph a line in slope-intercept form (y = mx + b)
• What a dashed line vs. solid line means
• Which side of the line to shade for > or <
• How to test a point to verify your shading

If any of those are fuzzy, fix that first. Otherwise, keep reading.

Graphing a Single Inequality: Quick Recap

For any inequality like y > 2x + 3:

• Convert to equals: y = 2x + 3
• Graph the boundary line
• Use a dashed line for > or <
• Use a solid line for ≥ or ≤
• Pick a test point (0,0) unless it falls on the line
• Shade the half-plane that makes the inequality true

Step-by-Step: Graphing a System of Inequalities

Example Problem

Graph this system:

y ≥ x - 2
y < -x + 4

Step 1: Graph Each Inequality One at a Time

Don't try to do both at once. Start with the first inequality.

For y ≥ x - 2:

• Boundary line: y = x - 2
• Solid line (because ≥)
• Test point (0,0): 0 ≥ 0 - 2 → 0 ≥ -2 ✓
• Shade above the line

Step 2: Graph the Second Inequality

For y < -x + 4:

• Boundary line: y = -x + 4
• Dashed line (because <)
• Test point (0,0): 0 < -0 + 4 → 0 < 4 ✓
• Shade below the line

Step 3: Find the Overlapping Region

The solution to your system is where the shading from both inequalities intersects. Look for the region that's shaded by BOTH graphs.

In this case, it's a wedge-shaped area bounded by the two lines, where they cross each other.

Step 4: Verify Your Answer

Pick a point in your overlapping region and plug it into both inequalities. If it works, you're correct. If not, your shading is wrong somewhere.

Common Mistakes That Will Mess You Up

How to Shade Multiple Inequalities Without Getting Confused

When you're working with more than two inequalities, shading gets messy fast. Here's what works:

• Use pencil first, then darken the final solution region
• Make each inequality's shading pattern different: one with horizontal lines, one with vertical, one with dots
• Use different colors if your teacher allows it
• The final answer is always the region covered by ALL patterns

Systems with More Than Two Inequalities

The process doesn't change. You just add more steps:

1. Graph inequality #1 and shade
2. Graph inequality #2 and shade (using a different pattern)
3. Graph inequality #3 and shade
4. Continue until all are graphed
5. The solution is wherever all shadings overlap

Three or four inequalities typically create a polygon (often a triangle or quadrilateral) as the solution region. That's normal.

Practice Problem: Try This One

Graph this system:

x ≥ 0
y ≥ 0
x + y ≤ 6

Think you can get it? The first two inequalities restrict you to the first quadrant. The third cuts off everything beyond the line x + y = 6. The solution is a right triangle in the first quadrant with vertices at (0,0), (6,0), and (0,6).

When the System Has No Solution

Sometimes inequalities don't overlap at all. If that's the case, the system has no solution.

This happens when the shaded regions are completely separate. You'll know it when you see it—there's literally no region that satisfies everything.

Quick Reference: Line Types and Shading

For shading direction: test (0,0) if possible. If the inequality is true at (0,0), shade toward (0,0). If false, shade away from (0,0).