Inequalities Solver
Solve linear and quadratic inequalities; output interval notation and a number-line sketch.
Input Data
Results
At a glance:For linear a·x+b (op) k: move terms, then x (op') (k−b)/a; flip the sign if a<0. For quadratic, set c'=c−k, find roots r₁≤r₂; a>0 opens up (positive outside roots), a<0 opens down (positive between roots), then choose the interval by the operator. Interval notation uses ( ) for excluded, [ ] for included; ∞ is always ( ).
Formula
Linear: x (op') (k−b)/a; flip when a<0.
Quadratic: c' = c − k; roots r₁, r₂ of a x²+b x+c'=0.
a>0: positive outside roots; a<0: positive between roots.
$$ax+b\;(\text{op})\;k$$$$ax^{2}+bx+c\;(\text{op})\;k$$How to Use
- Pick the degree (linear / quadratic).
- Enter a, b (and c for quadratic), the RHS k, and an operator.
- The tool returns interval notation, a number-line sketch and steps.
Quadratic solution (a>0)
| Op | Positive region | Solution |
|---|---|---|
| < 0 | between | (r₁, r₂) |
| ≤ 0 | between incl. | [r₁, r₂] |
| > 0 | outside | (−∞, r₁) ∪ (r₂, ∞) |
| ≥ 0 | outside incl. | (−∞, r₁] ∪ [r₂, ∞) |
For a<0 the positive/negative regions swap.
Case Studies
x² − 3x + 2 < 0
Roots 1, 2; a>0 ⇒ between: x ∈ (1, 2).
−2x < 4
Divide by −2, flip: x > −2.
FAQ
Why flip the sign for negative a?
Multiplying or dividing by a negative reverses the order, so < becomes > and ≤ becomes ≥.
Why is ∞ always a parenthesis?
∞ is not a real number and cannot be attained, so interval notation always uses ( ) for it.
No real roots?
If Δ<0 the quadratic is always positive (a>0) or always negative (a<0): pick all reals or no solution accordingly.
Related Tools
References
Content review: Calculatorism Science Team. Linear/quadratic sign-flip and interval notation have been verified. Results are for reference only.