Linear Inequality Solver
Solve linear inequalities of the form ax + b < c (or ≤, >, ≥) and express the answer as an interval or number line. Use it when solving algebraic inequalities in coursework or real-world constraint problems.
About this calculator
A linear inequality has the form ax + b ⋚ c, where a, b, and c are real numbers and ⋚ represents one of <, ≤, >, or ≥. Solving it is similar to solving a linear equation: isolate x by subtracting the constant from both sides and dividing by the coefficient. The critical rule is that dividing or multiplying both sides by a negative number reverses the inequality sign — for example, −2x > 6 becomes x < −3. The solution x = (c − b)/a gives the boundary point, and the inequality sign determines which side of that point is included in the solution set. Solutions are expressed as intervals: x < 3 becomes (−∞, 3), and x ≥ −1 becomes [−1, ∞). On a number line, open circles indicate strict inequalities (<, >) and closed circles indicate non-strict ones (≤, ≥).
How to use
Solve 3x − 4 > 8. Enter Coefficient = 3, Constant = −4 (so the left side is 3x + (−4)), Inequality Sign = >, Right Side = 8. The calculator computes x = (8 − (−4)) / 3 = 12 / 3 = 4. Because the coefficient 3 is positive, the sign does not flip, so the answer is x > 4, or in interval notation (4, ∞). Now try −2x + 1 ≤ 7: coefficient = −2, constant = 1, sign = ≤, right side = 7. Result: x = (7 − 1)/(−2) = −3, and because the coefficient is negative the sign flips, giving x ≥ −3, or [−3, ∞).
Frequently asked questions
Why do you flip the inequality sign when dividing by a negative number?
The inequality sign represents an ordering relationship on the number line. When you multiply or divide both sides of an inequality by a positive number, the relative order of the two sides is preserved. However, multiplying or dividing by a negative number reflects both values across zero on the number line, which reverses their order. For example, 4 > 2 is true, but multiplying both sides by −1 gives −4 and −2, and −4 < −2, so the sign must flip. Forgetting this rule is one of the most common algebra mistakes and produces solutions that are completely wrong.
How do you express a linear inequality solution in interval notation?
Interval notation uses parentheses for strict boundaries (the endpoint is not included, matching < or >) and square brackets for inclusive boundaries (the endpoint is included, matching ≤ or ≥). Infinity is always written with a parenthesis since it is not a real number that can be reached. For example, x > 5 becomes (5, ∞), x ≤ −2 becomes (−∞, −2], and −1 ≤ x < 4 becomes [−1, 4). This notation is compact and universally understood in algebra, calculus, and analysis, making it the preferred format in higher-level mathematics.
What is the difference between a linear inequality and a compound inequality?
A linear inequality involves a single condition on x, such as 2x + 3 > 7, producing one boundary and one half-line as the solution. A compound inequality combines two conditions using 'and' or 'or': an 'and' compound inequality like −1 < 2x + 3 ≤ 9 requires x to satisfy both conditions simultaneously, producing an interval between two boundary points. An 'or' compound inequality like x < −1 or x > 4 requires x to satisfy at least one condition, producing a union of two half-lines. Compound inequalities appear frequently in absolute value problems and in defining domains for real-world constraints such as acceptable temperature ranges or budget limits.