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.

Tutor-style math help

Graph Linear Inequalities: what to notice and how to work it

Inequalities skill

Inequalities describe a set of possible values. Solve the boundary like an equation, then decide which side of the boundary makes the statement true.

What to notice first

Watch the comparison sign from the first line to the last. Multiplying or dividing by a negative reverses the direction.

Common student mistake

Do not forget open and closed endpoints. Strict signs use open circles; signs with equals use closed circles.

Key formulas and cues

$a<b$

$a\le b$

$\text{multiply/divide by a negative} \Rightarrow \text{reverse the sign}$

$|x-a|<b \Rightarrow a-b<x<a+b$

A reliable path

Solve the boundaryTemporarily treat the inequality like an equation.
Choose the sideUse the sign or test a number if the direction is not obvious.
Graph the solutionUse the correct endpoint and shade the values that work.

Worked examples

Flip the sign

Example: $-3x>12$

Divide both sides by -3.
Reverse the inequality sign.
Simplify 12 divided by -3.

Answer: $x<-4$

Keep the sign

Example: $x+5\le9$

Subtract 5 from both sides.
No negative multiplication or division happened.
Keep the sign direction.

Answer: $x\le4$

Open pop-up practice

Try one before moving on

Try: Solve $-2x\le10$.

Answer: $x\ge-5$. Divide by -2 and flip the sign.

Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

Step-by-Step Solver Worksheet Maker ✏Practice Drills Flashcards

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

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 →

Effortless Math Team

2 weeks ago

Effortless Math Team

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