Inequality Calculator
Solve any linear inequality ax + b < c, or a compound inequality L < ax + b < U, with full step-by-step working, the solution set, and interval notation.
⚖️ What is an Inequality?
An inequality is a mathematical statement that compares two expressions using one of the signs < (less than), ≤ (less than or equal to), > (greater than), or ≥ (greater than or equal to). Unlike an equation, which asserts that two expressions are exactly equal, an inequality describes a range of values. A linear inequality such as 2x + 3 > 7 has the variable raised only to the first power, and its solution is a set of numbers rather than a single answer.
Inequalities appear constantly in real problems. A budget rule might say total spending must stay below a limit, x ≤ 5000. A speed limit sets v ≤ 60. A manufacturing tolerance requires a part width to fall inside a band, 9.8 ≤ w ≤ 10.2, which is a compound inequality. In optimisation and economics, systems of inequalities define the feasible region where every constraint is satisfied at once. Learning to solve them cleanly is a core algebra skill.
Solving a linear inequality is almost the same as solving a linear equation: you isolate the variable using inverse operations. There is one crucial extra rule. When you multiply or divide both sides by a negative number, the direction of the inequality reverses. Forgetting this flip is the most common error students make, so this calculator applies it automatically and shows exactly where it happens in the working.
This tool solves two families of problems. Linear mode handles a single inequality of the form ax + b compared with c. Compound mode handles a double inequality of the form L < ax + b < U, where the expression is trapped between a lower and an upper bound. Each result is given as a solution set, in interval notation, and with a full step-by-step explanation.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Basic Linear Inequality
Solve 2x + 3 > 7
Example 2 — Sign Flip with a Negative Coefficient
Solve −3x + 5 ≤ 14
Example 3 — Compound Inequality
Solve 1 ≤ 2x + 3 < 9
❓ Frequently Asked Questions
🔗 Related Calculators
How do you solve a linear inequality?
Solve it almost exactly like an equation: isolate the variable using inverse operations. For ax + b < c, subtract b from both sides to get ax < c − b, then divide by a. The one extra rule is that if a is negative, you must flip the inequality sign when you divide. For example, 2x + 3 > 7 gives 2x > 4, so x > 2.
When do you flip the inequality sign?
You flip the inequality sign whenever you multiply or divide both sides by a negative number. For example, −3x + 5 ≤ 14 becomes −3x ≤ 9, and dividing by −3 flips ≤ to ≥, giving x ≥ −3. Adding, subtracting, or dividing by a positive number never changes the direction of the inequality.
What is interval notation?
Interval notation describes a solution set using brackets. A round bracket ( or ) means the endpoint is excluded (strict < or >); a square bracket [ or ] means it is included (≤ or ≥). Infinity always uses a round bracket. So x > 2 is written (2, ∞), x ≤ 5 is (−∞, 5], and −1 ≤ x < 3 is [−1, 3).
What is a compound inequality?
A compound inequality combines two inequalities into one statement, such as 1 ≤ 2x + 3 < 9. It means the expression is simultaneously at least the lower bound and less than the upper bound. You solve it by performing each operation on all three parts at once: subtract 3 to get −2 ≤ 2x < 6, then divide by 2 to get −1 ≤ x < 3.
How is solving an inequality different from solving an equation?
The algebra is nearly identical, but there are two differences. First, an inequality usually has infinitely many solutions (a range) rather than one value. Second, multiplying or dividing by a negative number reverses the inequality sign, which never happens with equations. Otherwise the same inverse-operation steps apply.
What does 'no solution' or 'all real numbers' mean?
These occur when the coefficient a is zero, so the variable vanishes. If the remaining statement is true, such as 3 < 5, every value of x works and the answer is all real numbers, written (−∞, ∞). If it is false, such as 5 < 3, no value works and the answer is no solution, written with the empty-set symbol ∅.
Can inequalities have negative or fractional answers?
Yes. The boundary value (c − b) / a can be any real number, including negatives and fractions. For example, 4x + 1 < 0 gives x < −0.25. The calculator reports the exact decimal value and shows the same result in interval notation so you can read the boundary at a glance.
How do you graph an inequality solution on a number line?
Draw a number line, mark the boundary value, and use an open circle for a strict inequality (< or >) or a closed circle for an inclusive one (≤ or ≥). Then shade the ray in the direction of the solution: to the right for x greater than the boundary, to the left for x less than it. Compound solutions shade the segment between the two bounds.
Why does the compound inequality need the lower bound below the upper bound?
A compound inequality of the form L < expression < U only makes sense when L is less than or equal to U. If the lower bound were larger than the upper bound, no value could satisfy both parts at once, so the region would be empty. The calculator checks this and asks you to correct the bounds.