Graphing an inequality on a number line takes three decisions: where to place your point, whether to fill it in or leave it open, and which direction to shade. Once you know those three rules, you can handle any single or compound inequality you’ll see in a math class.
Open Circles vs. Closed Circles
The circle you draw at the boundary number tells the reader whether that number is part of the solution. A strict inequality (using < or >) means the variable can get close to that number but never equal it. You show this with an open circle, a small unfilled dot on the number line. For example, x > 4 gets an open circle on 4 because 4 itself is not a valid solution.
When the inequality includes “or equal to” (using ≤ or ≥), the boundary number is part of the solution. You show this with a closed circle, a filled-in dot. For x ≤ 10, you’d place a solid dot on 10 because 10 satisfies the inequality.
A quick reference:
- < or > → open circle (boundary not included)
- ≤ or ≥ → closed circle (boundary included)
Which Direction to Shade
After placing your circle, draw a line or arrow in the direction of all the values that make the inequality true. “Less than” means smaller numbers, which sit to the left on a standard number line. “Greater than” means larger numbers, which sit to the right.
Take T > 74. You’d place an open circle on 74, then draw an arrow extending to the right, covering 75, 76, 77, and every value beyond. For x ≤ 48, you’d place a closed circle on 48 and shade everything to the left, because every number smaller than 48 (plus 48 itself) is a solution. The arrow indicates the shading continues forever in that direction.
Step-by-Step Example
Suppose you need to graph x ≥ −2 on a number line.
- Step 1: Draw a number line and mark −2 on it.
- Step 2: Look at the symbol. The ≥ sign includes “equal to,” so draw a closed (filled) circle on −2.
- Step 3: The inequality reads “greater than or equal to,” so shade to the right of −2 and add an arrow showing the solutions continue toward positive infinity.
That’s the entire graph. Every point on the shaded region, including −2 itself, is a solution.
Solving Before Graphing
Sometimes the inequality isn’t in a ready-to-graph form like x > 4. You may need to isolate the variable first, and one rule trips up students more than any other: multiplying or dividing both sides by a negative number reverses the inequality sign. A < becomes >, and vice versa.
For example, solve −3x < 12. Divide both sides by −3 to isolate x, which flips the sign: x > −4. Now you can graph it with an open circle on −4 and shading to the right. If you forget to flip the sign, every value in your shaded region will be wrong. A good way to double-check is to pick a number from your shaded region and plug it back into the original inequality to confirm it works.
Compound Inequalities: “And” vs. “Or”
A compound inequality combines two conditions. How you graph it depends on the connecting word.
“And” Inequalities
An “and” inequality requires both conditions to be true at the same time, so the solution is the overlap (intersection) of the two sets. The classic example is 0 < x < 4, which is shorthand for x > 0 AND x < 4. On the number line, you place an open circle on 0, an open circle on 4, and shade only the region between them. Values outside that range fail at least one condition.
“Or” Inequalities
An “or” inequality requires at least one condition to be true, so the solution is the combined total (union) of both sets. For x < 3 OR x > 5, you place an open circle on 3 and shade to the left, then place an open circle on 5 and shade to the right. The middle section between 3 and 5 stays unshaded because neither condition is satisfied there.
The visual difference is easy to remember: “and” graphs typically shade inward (a bounded segment), while “or” graphs typically shade outward (two rays heading in opposite directions).
Reading a Graph as Interval Notation
Many classes ask you to convert a number line graph into interval notation. The rules map directly onto what you already know about circles and shading.
An open circle translates to a parenthesis, ( or ), meaning the endpoint is not included. A closed circle translates to a bracket, [ or ], meaning it is included. You always write the smaller number first, then a comma, then the larger number. If the shading continues forever to the right, use the infinity symbol ∞ with a parenthesis (never a bracket, because infinity is not a reachable number). If it continues forever to the left, use −∞ with a parenthesis.
Some examples to tie it all together:
- x > 4 → open circle on 4, shade right → (4, ∞)
- x ≤ 10 → closed circle on 10, shade left → (−∞, 10]
- 0 < x < 4 → open circles on both 0 and 4, shade between → (0, 4)
- x ≥ −2 → closed circle on −2, shade right → [−2, ∞)
Once you can move between the inequality, the number line graph, and the interval notation, you have a complete picture of the solution set in all three formats.