Graphing Inequalities Worksheet (Free PDF + Answer Key)

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Graphing Inequalities Worksheet: Free PDF, Worked Examples & Answer Key

✍ Written & fact-checked by Dr. Irfan Mansuri  |  Last Updated: August 2026

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By Dr. Irfan Mansuri
· August 2026
· 9 min read
Grades 6-9
Algebra 1
Graphing inequalities worksheet showing number line and coordinate plane examples with shaded regions and open closed circles
Graphing inequalities on a number line and coordinate plane — free printable worksheet with answer key

The first time I taught graphing inequalities, a student raised her hand halfway through my number-line demo and asked, “Wait — why is the circle sometimes hollow?” I had drawn the open circle, moved on, and assumed everyone understood. Half the class had not. That one question reshaped how I teach this topic: the circle type and the shading direction are the two places where students lose marks, and neither gets enough attention in most worksheets.

Graphing an inequality means representing all values that satisfy it as a visual region — a ray on a number line (one variable) or a shaded half-plane on a coordinate grid (two variables). The graph shows infinitely many solutions at once, which is why it is more powerful than a single numeric answer.

  • Understand when to use an open vs. closed circle on a number line.
  • Graph one-variable and two-variable linear inequalities correctly.
  • Know when to flip the inequality sign (and when NOT to).
  • Complete 10 graded practice problems and check your work with the answer key.
🎯 Learning Outcomes: After working through this page and the worksheet, you will be able to graph any one-variable or two-variable linear inequality accurately and explain your reasoning step by step.
⚡ Quick Answer: A graphing inequalities worksheet gives students practice plotting inequality solutions on a number line (one variable) or coordinate plane (two variables). Use an open circle for strict inequalities (< or >) and a closed circle for “or equal to” (≤ or ≥). For two-variable inequalities, graph the boundary line (dashed or solid), then shade the half-plane that satisfies the inequality using a test point.

📄 Free Printable PDF: 10 graded problems + full answer key, print-ready.
⬇ Download Free PDF Worksheet (with Answer Key)

📌 Quick Summary

  • Open circle = endpoint NOT included (< or >); closed circle = endpoint included (≤ or ≥).
  • Flip the inequality sign ONLY when multiplying or dividing by a negative number.
  • For two-variable inequalities: dashed line for strict, solid line for “or equal to.”
  • Use a test point (try (0,0) first) to decide which half-plane to shade.
  • The worksheet below has 10 problems, graded easy to hard, with a full answer key.
  • Compound inequalities graph as a segment or two rays — read the “and/or” carefully.

📊 Quick Facts

Topic Graphing Linear Inequalities
Grade Level Grades 6-9 (Algebra 1)
Skills Covered Number line, coordinate plane, compound inequalities
Problems in Worksheet 10 (Easy to Hard)
Answer Key Included (on-page + PDF)
Prerequisite Solving one-step and two-step equations
Common Error Rate ~60% of students flip the sign incorrectly at least once

What Is Graphing Inequalities?

Graphing an inequality means showing its solution set as a picture. Unlike an equation (which gives one or a few specific values), an inequality’s solution is usually an infinite set of values. A graph makes that infinite set visible and understandable at a glance.

There are two main types you will encounter in school:

Type Example Graph On What You Draw
One-Variable x > 3 Number line Open/closed circle + arrow
Two-Variable y ≤ 2x + 1 Coordinate plane Boundary line + shaded half-plane
Compound (and) -2 < x ≤ 5 Number line Segment between two endpoints
Compound (or) x < -1 or x ≥ 4 Number line Two separate rays

Graphing inequalities is a foundational algebra skill. It appears in systems of inequalities, linear programming, and real-world problems like budget constraints and speed limits. Mastering the basics here pays off in every math course that follows.

How to Graph Inequalities: Step-by-Step

Part A: One-Variable Inequalities on a Number Line

  1. Solve the inequality for the variable (if it is not already isolated).
  2. Locate the boundary value on the number line.
  3. Draw the correct circle: open (hollow) for < or >; closed (filled) for ≤ or ≥.
  4. Draw the arrow in the direction of the solution: right for > / ≥, left for < / ≤.
  5. Check: pick a value inside the shaded region and verify it satisfies the original inequality.

Part B: Two-Variable Inequalities on a Coordinate Plane

  1. Rearrange so y is alone on the left side (slope-intercept form helps).
  2. Graph the boundary line y = mx + b: dashed for < or >, solid for ≤ or ≥.
  3. Choose a test point not on the line. The origin (0, 0) is easiest unless the line passes through it.
  4. Substitute the test point into the original inequality.
  5. Shade: if the test point satisfies the inequality, shade its side; if not, shade the opposite side.
💡 Pro Tip: Always write the inequality sign direction as an arrow in your working. If you flip the sign (dividing by a negative), circle the flip in your working so you can check it later. This one habit eliminates most sign-flip errors.

Worked Examples (Study These Before Practising)

✏ Example 1 — One-Variable (Easy)

Graph x + 5 > 8

Step 1: Subtract 5 from both sides: x > 3.

Step 2: The boundary is 3. The sign is strict (>), so draw an open circle at 3.

Step 3: The arrow points right (greater than = larger values).

Check: Try x = 5: 5 + 5 = 10 > 8. ✓

📐 Visual — Number Line: x > 3
  <----+----+----+----O=============>
      0    1    2    3
                     ↑
               Open circle at 3
               Arrow goes RIGHT

✏ Example 2 — Sign Flip (Medium)

Solve and graph: -3x < 9

Step 1: Divide both sides by -3. Because we divide by a NEGATIVE, flip the sign: x > -3.

Step 2: Open circle at -3 (strict inequality), arrow pointing right.

Check: Try x = 0: -3(0) = 0 < 9. ✓ And x = -4: -3(-4) = 12, which is NOT < 9. ✓ (x = -4 should not be in the solution.)

✏ Example 3 — Two-Variable (Hard)

Graph y < 2x + 1 on a coordinate plane.

Step 1: The boundary line is y = 2x + 1. The sign is strict (<), so draw a dashed line.

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Step 2: Plot the line using slope 2 and y-intercept 1.

Step 3: Test point (0, 0): Is 0 < 2(0) + 1? Is 0 < 1? Yes. Shade the side containing (0, 0) — that is the region below the line.

📐 Visual — Coordinate Plane: y < 2x + 1
   y
   |          /  (dashed line)
 3 |        /
 2 |      /
 1 |----/---- (y-intercept)
   |  /
 0 |/ ← test point (0,0) is HERE
   +------------- x
   -1  0  1  2  3

  ░░░ = shaded region (below the dashed line)
🧑‍🏫 My POV

In my experience, the test-point method is underrated. Most textbooks show students to just “shade below for <” — but that shortcut only works when y is already isolated. The moment a student sees 3x – 2y ≤ 6 and tries to shade below the line, they get it wrong. Teaching the test-point method as the default approach, not the backup, eliminates this entire class of errors. I have seen test scores improve noticeably when students commit to this habit.

Common Mistakes — And How to Fix Them

❌ Mistake 1: Using a closed circle for a strict inequality.
Wrong: graphing x > 3 with a filled dot at 3.
✅ Fix: Ask yourself: “Is 3 itself a solution?” Substitute: 3 > 3 is false. So use an open circle.
❌ Mistake 2: Forgetting to flip the sign when dividing by a negative.
Wrong: -2x < 8 → x < -4.
✅ Fix: -2x < 8 → divide by -2, flip: x > -4. Always circle the negative divisor as a reminder.
❌ Mistake 3: Drawing a solid boundary line for a strict two-variable inequality.
Wrong: graphing y < 2x + 1 with a solid line.
✅ Fix: Strict signs (<, >) always get a dashed line. The dashed line means “not included.”
❌ Mistake 4: Shading the wrong half-plane without checking.
Wrong: assuming “less than” always means shade below.
✅ Fix: Always use a test point. It takes 10 seconds and is always correct.

🔑 The Part Most Guides Miss

The “shade below for <” rule is a trap for non-standard forms. Every worksheet site and most textbooks teach “if the sign is <, shade below the line.” This works perfectly when the inequality is already in the form y < mx + b. But consider 3x – 2y ≤ 6. If you shade below the line y = 1.5x – 3, you shade the wrong region, because rearranging gives y ≥ 1.5x – 3 — the shading is actually above the line.

The test-point method is the only approach that works for every form without rearranging first. Substitute (0, 0) into the original inequality as written. If it is true, shade the (0,0) side. If false, shade the other side. This single habit is worth more than memorising any shortcut.

Bonus non-obvious fact: When the boundary line passes through the origin, (0, 0) cannot be your test point. Use (0, 1) or (1, 0) instead. Most guides never mention this edge case, and it trips up students on tests.

📝 Graphing Inequalities — Practice Worksheet

Work through all 10 problems below. For problems 1-7, graph on a number line. For problems 8-10, graph on a coordinate plane. When you finish, check your answers using the key below.

How to use this worksheet: Print the PDF (button above or below) for a clean write-on version. Or work through the on-page list and sketch your graphs on paper. Use the collapsible answer key to self-check after each problem — do not peek until you have tried it yourself.

  1. Graph x > 3 on a number line.
  2. Graph x ≤ -2 on a number line.
  3. Graph -1 < x ≤ 4 on a number line.
  4. Graph x ≥ 0 on a number line.
  5. Solve and graph: x + 5 > 8
  6. Solve and graph: 2x ≤ 10
  7. Solve and graph: -3x < 9
  8. Graph y < 2x + 1 on a coordinate plane.
  9. Graph y ≥ -x + 3 on a coordinate plane.
  10. Graph 3x – 2y ≤ 6 on a coordinate plane.
✅ Show Answer Key
  1. Open circle at 3, arrow pointing right.
  2. Closed circle at -2, arrow pointing left.
  3. Open circle at -1, closed circle at 4, segment connecting them.
  4. Closed circle at 0, arrow pointing right.
  5. x > 3. Open circle at 3, arrow pointing right.
  6. x ≤ 5. Closed circle at 5, arrow pointing left.
  7. x > -3 (flip sign — divided by negative). Open circle at -3, arrow pointing right.
  8. Dashed boundary line y = 2x + 1; shade the region below the line (test: (0,0) gives 0 < 1, true).
  9. Solid boundary line y = -x + 3; shade the region above the line (test: (0,0) gives 0 ≥ 3, false — shade opposite side, which is above).
  10. Rearrange: y ≥ 1.5x – 3. Solid boundary line y = 1.5x – 3; shade above the line (test: (0,0) gives 0 ≥ -3, true — shade the (0,0) side, which is above).

📄 Get the printable version: Clean PDF with blank graphs, answer key on a separate page.
⬇ Download Free PDF Worksheet (with Answer Key)

🔍 Reveal-on-Click Practice Problems

Try each problem, then click to reveal the full worked solution.

Problem A: Solve and graph 4x – 3 ≥ 9
Step 1: Add 3 to both sides: 4x ≥ 12.
Step 2: Divide by 4 (positive — no flip): x ≥ 3.
Graph: Closed circle at 3, arrow pointing right.
Problem B: Solve and graph -5x + 2 > 17
Step 1: Subtract 2: -5x > 15.
Step 2: Divide by -5 — FLIP the sign: x < -3.
Graph: Open circle at -3, arrow pointing left.
Problem C: Graph y > -2x + 4 on a coordinate plane
Step 1: Boundary line is y = -2x + 4. Sign is strict (>) → dashed line.
Step 2: Test (0, 0): Is 0 > -2(0) + 4? Is 0 > 4? No.
Step 3: Shade the side that does NOT contain (0, 0) — the region above the line.

📐 Also Useful: Complete Geometry Formula Sheet →

🧠 Quick Quiz — Test Yourself

Q1. What circle do you use to graph x < 5 on a number line?



See Answer
✅ B) Open circle. The sign is strict (<), so 5 is NOT included. Open circle at 5, arrow pointing left.

Q2. You solve -4x > 20. What is the correct answer?



See Answer
✅ B) x < -5. Dividing by -4 (negative) flips the sign. -4x > 20 → x < -5.

Q3. For y ≤ 3x – 2, the boundary line should be:



See Answer
✅ B) Solid. The “or equal to” (≤) means points on the boundary line ARE solutions, so the line is solid.

❓ Frequently Asked Questions

What Is the Difference Between an Open Circle and a Closed Circle on a Number Line?
An open circle means the endpoint is NOT included in the solution — used for strict inequalities (< or >). A closed (filled) circle means the endpoint IS included — used for ≤ or ≥. A quick test: substitute the boundary value into the inequality. If it makes the inequality true, use a closed circle. If false, use an open circle.
When Do You Flip the Inequality Sign?
You flip the inequality sign only when you multiply or divide both sides by a negative number. Adding or subtracting any number — even a negative one — never flips the sign. This is the single most tested rule in inequality problems. A helpful memory trick: think of the number line flipping direction when you multiply by -1.
How Do You Know Which Side to Shade on a Coordinate Plane?
Pick a test point not on the boundary line — the origin (0, 0) is the easiest choice unless the line passes through it. Substitute the test point into the original inequality. If the result is true, shade the side containing that test point. If false, shade the opposite side. This method works for every inequality form, including non-standard ones like 3x – 2y ≤ 6.
Is the Boundary Line Solid or Dashed?
The boundary line is dashed for strict inequalities (< or >) because points on the line are not part of the solution set. It is solid for ≤ or ≥ because those points are included. A simple rule: if you would use an open circle on a number line, use a dashed line on a coordinate plane. If you would use a closed circle, use a solid line.
What Grade Level Is Graphing Inequalities?
One-variable inequalities on a number line are typically introduced in grades 6-7 as part of pre-algebra. Two-variable linear inequalities on a coordinate plane are a standard Algebra 1 topic, usually covered in grades 8-9. Compound inequalities and systems of inequalities extend the skill into Algebra 2.
How Is Graphing an Inequality Different from Graphing an Equation?
Graphing an equation gives a single line or curve — the exact set of points that make the equation true. Graphing an inequality gives a boundary line plus an entire shaded region, representing infinitely many solutions. The shaded half-plane is the defining feature of an inequality graph. On a number line, an equation gives a single point; an inequality gives a ray or segment.
Can an Inequality Have No Solution?
Yes.

Sources & References

Written and fact-checked by Dr Irfan Mansuri. External links open in a new tab and are provided for further reading and verification.

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