An open circle on a graph means that a specific point is not included. It’s a hollow dot, visually distinct from a filled-in (closed) circle, and it signals “up to here, but not this value.” You’ll encounter open circles most often in two situations: graphing inequalities on a number line and showing holes or endpoints in function graphs.
Open Circles on Number Lines
When you graph an inequality like x < 5, you place an open circle at 5 and draw a line or arrow extending to the left. The open circle tells you that 5 itself is not part of the solution. Every number less than 5 works, but 5 does not. The same logic applies to “greater than” inequalities: x > 3 gets an open circle at 3 with an arrow pointing right.
This is strictly tied to the symbols < (less than) and > (greater than). If the inequality uses ≤ (less than or equal to) or ≥ (greater than or equal to), you use a closed circle instead, because the boundary value is included in the solution set. So the visual shorthand is simple: hollow means excluded, filled means included.
Open vs. Closed Circles at a Glance
- Open circle (hollow): The value at that point is not included. Used with strict inequalities (< or >).
- Closed circle (filled): The value at that point is included. Used with non-strict inequalities (≤ or ≥).
If you see x < −1 on a number line, the open circle sits at −1. All numbers to the left of −1 are solutions, but −1 itself is not. Swap the symbol to x ≤ −1, and that circle fills in, meaning −1 now counts as a valid answer too.
Open Circles on Function Graphs
Open circles also appear on coordinate plane graphs, not just number lines. Here they serve a slightly different but related purpose: they mark a point where the function doesn’t have a value, even though the curve seems like it should pass through that spot.
Holes in Curves
Sometimes a function is defined everywhere along a smooth curve except at one specific x-value. This creates what’s called a removable discontinuity, or simply a “hole.” On the graph it looks like a continuous line with a single open circle punched out of it. For example, the function f(x) = (x+2)(x+1) / (x+1) looks exactly like the straight line y = x + 2, except at x = −1. At that point, the numerator and denominator both equal zero, so the function is undefined. The graph shows the line passing through that spot with a hollow circle at (−1, 1) to mark the gap.
The curve approaches that point from both sides and would connect perfectly if the hole weren’t there. That’s what makes it “removable.” You could redefine the function at that single point to fill the gap, and the graph would become a complete, unbroken line.
Piecewise Function Endpoints
Piecewise functions, which follow different rules over different intervals, use open and closed circles to show where each piece starts and stops. If a piece of the function covers all x-values up to but not including 2, there’s an open circle at x = 2. The next piece might pick up at x = 2 with a closed circle, showing that this second rule is the one that actually defines the function’s value at that point. When one piece ends with an open circle and the next begins with a closed circle at a different height, you can visually see the “jump” between them.
Why the Notation Matters
The difference between an open and closed circle can change an answer. On a number line, including or excluding a boundary point changes which values satisfy an inequality. On a function graph, a hole means the function literally has no output at that x-value, which affects calculations involving that point. If you’re reading a graph and you see a hollow dot, the takeaway is always the same: that exact coordinate is excluded. The graph is telling you “the pattern goes here, but this specific point does not belong.”
In some contexts outside pure math, like scatter plots or data charts, hollow markers can serve a different purpose entirely, such as distinguishing between two data sets or categories. But in algebra, precalculus, and calculus courses, the convention is consistent: an open circle means “not included.”