Graphing Systems of Linear Inequalities- Complete Tutorial

What Is a System of Linear Inequalities?

A system of linear inequalities is just two or more linear inequalities grouped together. You're not finding a single point anymore—you're finding the entire region where all inequalities are true at the same time.

Think of it like this: each inequality is a rule. The solution to the system is everywhere that all rules apply simultaneously. That's the overlap. That's what you need to find.

What You Must Know Before You Start

Don't skip this part. If you can't graph a single linear inequality, you can't graph a system. Full stop.

Key prerequisites:

If any of this is fuzzy, go back and review. This isn't the place to patch gaps.

How to Graph a System of Linear Inequalities

Here's the actual process. No fluff.

Step 1: Graph Each Inequality Separately

Handle one inequality at a time. Don't try to do everything at once.

For y ≤ 2x + 3:

For y > -x + 1:

Step 2: Identify the Overlap

After you've graphed both inequalities, look for where the shaded regions intersect. That's your solution region.

The solution to the system is every point that satisfies both inequalities. If a point is in one shaded area but not the other, it doesn't count.

Step 3: Check a Test Point

Pick any point in the overlap. Plug it into both inequalities. If it works, you're probably correct. If it doesn't, you shaded the wrong side on at least one inequality.

Solid Line vs. Dashed Line: The Rule

This trips people up constantly. Here's the distinction:

It matters. A solid line means points ON the line are solutions. A dashed line means they aren't.

Common Mistakes That Will Mess You Up

Shading the wrong direction

This is the number one error. For y >, shade above. For y <, shade below. If you mix this up, your entire solution region is wrong.

Forgetting to check both inequalities

Each inequality defines its own region. You need both. Don't graph one and call it done.

Mixing up ≤ and <

The difference between a solid and dashed line is critical. ≤ means solid. < means dashed. Same pattern for ≥ versus >.

Not using the test point

Test points catch errors. Use them. Pick (0,0) if it's not on any boundary line—it's the easiest point to check.

Quick Reference Table

Symbol Line Type Shade Direction Boundary Included?
> Dashed Above No
≥ Solid Above Yes
< Dashed Below No
≤ Solid Below Yes

Practical Example: Putting It All Together

Graph this system:

y ≥ x - 2
y < -2x + 4

First inequality: y ≥ x - 2

Second inequality: y < -2x + 4

The solution is the region where both conditions are met—above the first line AND below the second line. The overlap is your answer.

What If There's No Overlap?

Sometimes the shaded regions don't intersect. When that happens, the system has no solution.

This happens when the inequalities describe regions that can't coexist. For example:

Nothing is simultaneously greater than 5 and less than 3. When you graph this, you'll see two shaded regions pointing away from each other. No overlap means no solution.

When the System Has Infinite Solutions

If both inequalities are identical, you get one shaded region. The solution is every point that satisfies that one inequality—which is infinite.

More commonly, you'll see bounded regions—areas where the overlap creates a closed shape, like a triangle or quadrilateral. Every point inside that shape is a solution.

How to Check Your Work

Pick a point in the overlap. Any point.

Test it in both original inequalities. If both check out, you're correct. If one fails, you made an error—usually shading the wrong direction on one of the lines.

Test point: (0, 0) works if neither line passes through the origin. It's the fastest check you can do.

Bottom Line

Graph each inequality one at a time. Use solid lines for ≤ and ≥, dashed for < and >. Shade correctly—above for > and ≥, below for < and ≤. Find where the shading overlaps. That's your solution.

It's not complicated. It just requires precision. Get the basics right and the rest takes care of itself.