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.

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.

Worked Examples (Study These Before Practising)

  <----+----+----+----O=============>
      0    1    2    3
                     ↑
               Open circle at 3
               Arrow goes RIGHT
  

   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)
  

Common Mistakes — And How to Fix Them

📝 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.