Algebra
Single-variable inequalities
Graphing inequalities
20 practice questions
2 video lessons
Theory + worked examples
Theory
Graph a one-variable inequality on a number line:
- Open circle for \(<\) or \(>\) (endpoint not included).
- Closed circle for \(\le\) or \(\ge\) (endpoint included).
- Arrow toward all the values in the solution.
The circle shows whether the boundary is included; the arrow shows the direction.
\(x\ge2\): closed circle, arrow right.
Open vs closed circles.
Circle rules:
\[<,>\ \Rightarrow\ \text{open};\qquad \le,\ge\ \Rightarrow\ \text{closed}\]
Arrow right for greater, left for less.
How to graph
- Solve for the variable.
- Mark the boundary with an open or closed circle.
- Draw an arrow toward the solution set.
- Read a graph by reversing these rules.
Example 1 — Closed circle
Graph \(x\ge2\).
Solution
\(\ge\) uses a closed circle at \(2\), arrow right.
| \(x\ge2\) | \(\Rightarrow\) | \(\bullet\!\rightarrow\) |
Example 2 — Open circle
Graph \(x<-1\).
Solution
\(<\) uses an open circle at \(-1\), arrow left.
| \(x<-1\) | \(\Rightarrow\) | \(\circ\!\leftarrow\) |
Example 3 — Solve then graph
Solve and graph \(2x\le8\).
Solution
Divide by \(2\), then graph.
| \(x\) | \(\le\) | \(4\) |
Closed circle at \(4\), arrow left.
Example 4 — Read a graph
An open circle at \(3\) with an arrow right represents what?
Solution
Open means not included; arrow right means greater.
| \(x>3\) |
Common pitfalls
Open circle for strict \(<,>\); closed for \(\le,\ge\).
Arrow direction matches greater (right) or less (left).
Shade toward all solutions, not just the boundary.
Frequently asked questions
When do you use an open circle?
For strict inequalities \(<\) and \(>\).
When do you use a closed circle?
For \(\le\) and \(\ge\) — the endpoint is included.
Which way does the arrow point?
Toward the values that satisfy the inequality.
What does a closed circle at 2 with a left arrow mean?
\(x\le2\).
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