Inequality Calculator

Solve linear inequalities: ax + b < c, ax + b ≤ c, etc. Find the solution set, write in interval notation, and understand when the inequality sign flips.

∑ Algebra📐 ax + b < c → x < (c-b)/a🔢 Math
Coefficient a
Constant b
Right side c
Inequality sign
Inequality
Please enter valid values.

Formula & Reference

VariableSymbolFormulaUnits
Inequality Calculator—ax + b < c → x < (c-b)/asolution set

Step-by-Step Examples

Example 1
Simple

3x − 6 < 9.

  • 3x < 15
  • x < 5
  • Interval: (−∞, 5)
✓ x < 5
Example 2
Sign Flips

−2x + 4 ≥ 10.

  • −2x ≥ 6
  • Divide by −2 (flip sign)
  • x ≤ −3
  • Interval: (−∞, −3]
✓ x ≤ −3
Example 3
Greater Than

5x − 1 > 14.

  • 5x > 15
  • x > 3
  • Interval: (3, ∞)
✓ x > 3

Real-World Applications

📚
Education
Inequalities appear in every algebra course and standardized test.
💼
Economics
Budget constraints, feasibility regions in linear programming.
🧪
Science
Physical bounds: speed must be > 0, temperature must be < max.
📊
Data Science
Threshold filtering: find all records where value > threshold.

Common Mistakes to Avoid

⚠️
Forgetting to flip sign when dividing by negative

The most common inequality mistake. −2x < 6 → x > −3, NOT x < −3.

⚠️
Open vs closed brackets in interval notation

< and > use parentheses (excludes endpoint). ≤ and ≥ use brackets [includes endpoint].

⚠️
Multiplying both sides by negative expression

If multiplying by a variable expression, you don't know the sign — this is why we don't multiply by variables in inequalities.

Frequently Asked Questions

What is an inequality? ▾
A statement that one expression is greater than, less than, or equal/not equal to another. Solution is a range of values, not a single point.
When does the inequality sign flip? ▾
When you multiply or divide both sides by a negative number. NOT when adding/subtracting, and NOT when multiplying by a positive number.
What is interval notation? ▾
A way to write solution sets: (a,b) means a < x < b (open). [a,b] means a ≤ x ≤ b (closed). (−∞,5] means x ≤ 5.
How to solve compound inequalities? ▾
Type AND (like 2 < x < 5): solve both and intersect. Type OR (like x < 1 or x > 4): solve both and union.
What is a linear programming problem? ▾
Optimizing (maximizing or minimizing) a linear objective function subject to linear inequality constraints. Feasible region determined by inequalities.

Related Math Calculators

∑ Linear Equation∑ Absolute Value∑ Systems of Equations∑ Quadratic∑ Completing the Square

Formula Explorer connections

Interpretation: This formula describes an algebraic relationship among variables, functions, equations, roots or sequences. Assumption: Respect the expression’s domain and excluded values. Check roots in the original equation because transformations can introduce extraneous solutions.

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