Linear inequality
Solve a linear inequality with the solution shown on a number line.
Enter the coefficients and pick a type – the solution and number line appear instantly.
How to solve a linear inequality
A linear inequality is solved much like an equation – isolate x. There is one crucial rule though: when you divide (or multiply) an inequality by a negative number, the inequality sign flips (> becomes < and vice versa).
ax + b > 0 ⟹ ax > −b ⟹ x > −b/a (for a > 0)
Example: 2x − 4 > 0 → 2x > 4 → x > 2, i.e. x ∈ (2, +∞). If a were negative, e.g. −2x − 4 > 0 → −2x > 4 → x < −2 (the sign flipped).
Interval notation
The solution of an inequality is usually an interval. A round bracket means the boundary point does not belong to the solution (for strict > <), a square bracket means it does belong (for ≥ ≤). For example x > 2 is (2, +∞), while x ≥ 2 is [2, +∞).
Special cases (a = 0)
When a = 0, x disappears from the inequality and what remains is just a statement about numbers (e.g. −4 > 0). This either holds for all x (the solution is ℝ), or never holds (the solution is the empty set). The calculator recognises both cases.
Tip: an empty circle on the number line means the boundary does not belong to the solution, a filled one means it does.