How to Graph Linear Inequalities- Complete Tutorial

What Linear Inequalities Actually Are

Linear inequalities look like linear equations, but instead of an equals sign, you get one of these symbols: <, >, ≤, or ≥. That's the whole difference.

When you graph a linear equation like y = 2x + 3, you get a single line. When you graph the inequality y > 2x + 3, you get a region—a whole half-plane of points that satisfy the condition.

That's the key distinction: an equation gives you a line, an inequality gives you an area.

The Boundary Line: Your Starting Point

Every linear inequality has a boundary line. You find it by temporarily replacing the inequality symbol with an equals sign.

For y > 2x + 3, the boundary line is y = 2x + 3.

For 3x - 4y ≤ 12, the boundary line is 3x - 4y = 12.

Graph this line first. Everything else depends on it.

Dashed vs. Solid Lines

This trips people up constantly:

• Use a dashed line when the inequality is strict: < or >. The boundary is not included in the solution.
• Use a solid line when the inequality includes equality: ≤ or ≥. The boundary is included.

It's a small detail that changes everything about how your graph looks.

Which Side to Shade

Now you need to decide which half-plane contains your solution. Two methods work here.

Method 1: The Test Point

Pick a point that's not on the boundary line. The origin (0, 0) is usually the easiest.

Plug those coordinates into the original inequality. If the statement comes out true, shade that side. If it's false, shade the opposite side.

Example: y > 2x + 3

Test (0, 0): 0 > 2(0) + 3 → 0 > 3

That's false. So shade the side that doesn't contain (0, 0).

Method 2: Y-Intercept Logic

For inequalities in slope-intercept form (y > mx + b), you can use logic:

• y > mx + b means shade above the line
• y < mx + b means shade below the line

This only works reliably when y is already isolated. When it's not, use the test point method instead.

Step-by-Step: How to Graph Linear Inequalities

Let's walk through 2x + y ≤ 4 as a complete example.

Step 1: Convert to equality

Replace ≤ with =. You get 2x + y = 4.

Step 2: Solve for y

Get y by itself: y = -2x + 4.

Step 3: Graph the boundary line

Since the original uses ≤ (includes equality), draw a solid line for y = -2x + 4.

Step 4: Test a point

Use (0, 0): 2(0) + 0 ≤ 4 → 0 ≤ 4

True. Shade the side containing (0, 0).

That's it. Solid line, shaded region on one side, you're done.

What the Symbols Actually Mean

Common Mistakes to Avoid

Dashing when you should be solid. The ≤ and ≥ symbols include equality. Forgetting this makes your graph technically wrong, even if the shading looks right.

Shading the wrong side. Test point errors are the number one reason people get these problems wrong. Always verify with (0, 0) or another simple point.

Forgetting to solve for y first. Graphing 3x - 2y > 6 directly is painful. Isolate y first: -2y > -3x + 6, then y < 1.5x - 3. Much easier.

Drawing the line wrong. This sounds obvious, but slope errors happen. Check your y-intercept and your direction before moving on to shading.

Graphing Inequalities with Two Variables on One Axis

Sometimes you'll see something like x ≤ 3. This is still a linear inequality.

The boundary is the vertical line x = 3. Since it uses ≤, draw a solid line.

Shade everything to the left, where x is less than or equal to 3.

The same logic applies to horizontal boundaries like y > -2. Solid or dashed line at y = -2, shade above or below depending on the sign.

Quick Practice Tips

• Always convert to slope-intercept form first. It makes everything clearer.
• Get in the habit of testing (0, 0) immediately after drawing the line.
• When in doubt about shading direction, test a point. The test point never lies.
• Practice with ≤ and ≥ until the solid/dashed distinction is automatic.

Graphing linear inequalities comes down to three things: draw the line correctly, pick the right line style, and shade the correct side. That's the whole process. Once you internalize those steps, you'll never struggle with them again.