How to Graph Linear Inequalities? (+FREE Worksheet!)

Algebra 1

How to Graph Linear Inequalities

Graphing a linear inequality is graphing a line, then shading the side that makes it true. Two small decisions — dashed or solid boundary, and which side to shade — are the whole game. We’ll nail both with a test point. Solver, drills, and a worksheet maker a tap away.

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Illustration of students learning How to Graph Linear Inequalities

To graph a linear inequality, you do two things: draw the boundary line, then shade the side of the plane that makes the inequality true. Every point in the shaded region is a solution — which means a linear inequality has infinitely many solutions, a whole region of them. Two small choices decide everything, and a single “test point” settles both. Get these two right and every inequality graph works the same way.

The big idea

Boundary Line + Shaded Region

A linear inequality like $y < 2x - 1$ splits the coordinate plane into two halves with a boundary line. One half makes the inequality true; you shade it.

How to graph a linear inequality (3 steps):

Graph the boundary line $y = mx + b$. Use a dashed line for $<$ or $>$ (boundary not included) and a solid line for $\le$ or $\ge$ (boundary included).
Pick a test point not on the line — $(0,0)$ is easiest when the line doesn’t pass through it.
If the test point makes the inequality true, shade its side; if false, shade the other side.

Worked on the grid

Graphing $y < 2x - 1$

The boundary $y = 2x – 1$ is dashed (strict $<$). Test $(0,0)$: is $0 < 2(0)-1 = -1$? No — so $(0,0)$ is not in the region, and we shade the other side, below the line (away from the origin). The shaded half is the solution.

⚡ Graph an inequality

The Two Decisions

Decision 1

Dashed or solid?

Is the boundary itself included?

$<$ or $>$ → dashed (not included).
$\le$ or $\ge$ → solid (included).

Decision 2

Which side?

Let a test point decide.

Plug $(0,0)$ in. True → shade that side. False → shade the other.

Quick check

Predict, then verify

When it’s solved for $y$:

$y >$ usually shades above, $y <$ below — use this to predict, then confirm with a test point.

Worked Examples

A. Strict — dashed line

Graph $y < 2x - 1$.

Dashed boundary (strict). Test $(0,0)$: $0 < -1$ is false, so shade the side away from the origin. Dashed line, shade below the line.

B. Inclusive — solid line

Graph $y \ge -x + 3$ (see graph below).

Solid boundary ($\ge$). Test $(0,0)$: is $0 \ge -0+3 = 3$? No — the origin is out, so shade the other side (above the line). Solid line, shade above.

C. Through the origin

Graph $y > x$.

The line passes through $(0,0)$, so test a different point like $(0,2)$: $2 > 0$ is true. Dashed boundary, shade above the line. Pick a test point off the line.

D. A vertical boundary

Graph $x \le 3$.

The boundary is the vertical line $x=3$, drawn solid ($\le$). Test $(0,0)$: $0 \le 3$ is true, so shade everything to the left. Here a test point is the only reliable method.

Where This Shows Up

Regions show up whenever there’s a limit with two variables. If hot dogs cost $2 and drinks $3 and you have at most $12, then $2x + 3y \le 12$ — and every point in the shaded region is a combination you can afford. Businesses use these shaded “feasible regions” to plan production within budget and time limits. The graph turns a money rule into a picture of every choice that works.

Easy Points to Lose

Wrong boundary style. Strict ($<$, $>$) is dashed; inclusive ($\le$, $\ge$) is solid. Mixing these up is the most common deduction.
Shading the wrong side. Always test a point. Don’t guess from the sign alone — especially if the inequality isn’t solved for $y$.
Testing a point on the line. The test point must be off the boundary, or it can’t tell you anything. Pick $(0,0)$ unless the line goes through it.
Forgetting to flip when solving for $y$. If you divide by a negative to isolate $y$, flip the inequality first — then graph.

Your Turn: Boundary and Shading

For each inequality, state whether the boundary is dashed or solid and whether $(0,0)$ is in the solution. Reveal to check.

$y \le x + 1$
$y > 2x$
$y < -x + 2$

Show answers

$\color{blue}{\text{solid; } (0,0): 0\le 1 \text{ true → shade the side with the origin}}$
$\color{blue}{\text{dashed; } (0,0)\text{ on line, test }(0,1): 1>0 \text{ true → shade above}}$
$\color{blue}{\text{dashed; } (0,0): 0<2 \text{ true → shade the side with the origin}}$

Keep practicing

Make Your Own Inequality-Graphing Worksheet

Generate fresh inequalities to graph, with a full answer key — print or save as a PDF.

New problems every click — never the same sheet twice

Step-by-step answer key so you can self-check

📄 Open the Worksheet Maker ✏️ Quick Practice Drills

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Frequently Asked Questions

When is the boundary line dashed vs. solid?

Dashed for strict inequalities ($<$ or $>$) because the points on the line are not solutions. Solid for $\le$ or $\ge$ because the boundary points are included.

How do I know which side to shade?

Pick a test point that isn’t on the line (usually $(0,0)$) and plug it in. If the inequality is true, shade that point’s side; if false, shade the opposite side.

What if the boundary passes through the origin?

Then $(0,0)$ is on the line and can’t be a test point. Choose any other off-line point, like $(0,1)$, and test that instead.

How is this different from graphing a line?

You still graph the line, but you also decide dashed vs. solid and shade a whole half-plane. The solution is a region of points, not just the line.

Continue Your Study

Ready for the next step? Pick up right where this lesson leaves off:

Up nextFinding the x-InterceptContinue with Linear FunctionsOpen lesson →Up nextFinding the y-InterceptContinue with Linear FunctionsOpen lesson →Up nextWrite an Equation from a GraphContinue with Linear FunctionsOpen lesson →

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