Understanding the sign-flip rule
The sign of the coefficient a determines whether the inequality symbol stays the same or reverses when isolating x, as shown below.
| Sign of a | Effect on inequality symbol | Example |
|---|---|---|
| a > 0 (positive) | Symbol unchanged | 2x < 8 → x < 4 |
| a < 0 (negative) | Symbol reverses (< becomes >, ≤ becomes ≥, etc.) | −2x < 8 → x > −4 |
| a = 0 | Reduces to a true/false statement about constants only | 0x + 3 < 11 → always true (all real numbers) |
- When a = 0, the result is either 'All real numbers (always true)' or 'No solution (always false)', because the inequality no longer depends on x at all.
- The boundary value shown is the point at which the linear expression ax + b exactly equals c; whether that boundary itself is included in the solution depends on whether the inequality is strict (<, >) or inclusive (≤, ≥).
- This calculator solves one-variable linear inequalities only. Compound inequalities (e.g. 3 < 2x + 1 < 9) and inequalities involving x² or higher powers require different solution methods.
What is a linear inequality?
A linear inequality is a mathematical statement comparing a linear expression (a variable raised only to the first power, with a coefficient and constant) to another value using an inequality symbol: less than (<), less than or equal to (≤), greater than (>), or greater than or equal to (≥). Unlike a linear equation, which typically has one solution, a linear inequality usually has infinitely many solutions — every value of x that makes the statement true.
This calculator solves inequalities in the form ax + b [operator] c, where a is the coefficient of x, b is a constant added on the left side, and c is the value being compared against on the right. Solving means isolating x on one side, producing a result such as x < 4, meaning every value less than 4 satisfies the original inequality.
The single most important rule that distinguishes solving inequalities from solving equations is this: multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality symbol. This calculator applies that rule automatically based on the sign of the coefficient a.
How to use this inequality calculator
- Enter the coefficient of x (a) — the number multiplying x on the left side of the inequality.
- Enter the constant (b) — the number added to ax on the left side. Use 0 if there is no constant term.
- Select the inequality symbol: <, ≤, >, or ≥.
- Enter the value on the right side (c). Read the solution, expressed as x [symbol] boundary value; the sign flips automatically if a is negative.
The method for solving linear inequalities
Solving ax + b [op] c for x follows the same algebraic steps as solving an equation: subtract b from both sides, then divide both sides by a. Worked example: solve 2x + 3 < 11. Subtract 3 from both sides: 2x < 8. Divide both sides by 2 (a positive number, so the inequality direction is unchanged): x < 4.
When the coefficient a is negative, dividing both sides by it reverses the inequality symbol. Worked example: solve −2x + 3 < 11. Subtract 3: −2x < 8. Divide both sides by −2 and flip the symbol: x > −4 (not x < −4). This sign-flip rule is the single most common source of errors when solving inequalities by hand.
A special case arises when a = 0: the inequality reduces to a constant comparison, such as 3 < 11, which is either always true (in which case the solution is 'all real numbers') or always false (in which case there is 'no solution'), independent of x.
Common mistakes
- Forgetting to flip the inequality symbol when multiplying or dividing both sides by a negative number — this is the single most common error when solving inequalities.
- Treating ≤ and < (or ≥ and >) as interchangeable — the distinction matters for whether the boundary value itself is included in the solution set.
- Applying equation-solving intuition without checking the sign of the coefficient — an inequality with a negative coefficient does not solve the same way as a positive one.
- Assuming an inequality always has a numeric boundary — when the coefficient of x is 0, the result collapses to 'always true' or 'always false' with no boundary value at all.
अक्सर पूछे जाने वाले सवाल
How do you solve a linear inequality?
Isolate the variable using the same steps as solving an equation — add or subtract to move constants, then multiply or divide to remove the coefficient — but reverse the inequality symbol whenever you multiply or divide both sides by a negative number. For 2x + 3 < 11: subtract 3 to get 2x < 8, then divide by 2 to get x < 4.
Why does the inequality sign flip when multiplying by a negative number?
Multiplying or dividing by a negative number reverses the order of values on the number line — for example, 2 < 5, but multiplying both sides by −1 gives −2 and −5, and −2 is actually greater than −5. To keep the statement true, the inequality symbol must reverse: −2 > −5.
What does 'no solution' mean for an inequality?
'No solution' occurs when the coefficient of x is zero and the resulting constant comparison is false — for example, 0x + 3 > 11 simplifies to 3 > 11, which is never true regardless of x. In this case, no value of x can satisfy the original inequality.
What is the difference between < and ≤?
< (strictly less than) excludes the boundary value itself from the solution, while ≤ (less than or equal to) includes it. For x < 4, the value 4 is not a solution; for x ≤ 4, the value 4 is a solution. This distinction is visually represented on a number line by an open circle (<, >) versus a closed/filled circle (≤, ≥) at the boundary.
How do you check the solution to an inequality?
Substitute a test value from your solution range back into the original inequality and confirm it holds true, and optionally test a value outside the range to confirm it does not. For the solution x < 4 to 2x + 3 < 11, testing x = 0 gives 2(0) + 3 = 3 < 11 ✓, while testing x = 5 (outside the range) gives 2(5) + 3 = 13, which is not < 11, confirming the boundary is correct.
संदर्भ
- NIST Digital Library of Mathematical Functions (DLMF), §1.2 Elementary Algebra: Inequalities. dlmf.nist.gov.
- Sullivan M. Algebra and Trigonometry. 11th ed. Pearson, 2019. (Linear inequalities in one variable.)
- Lial ML, Hornsby J, Schneider DI, Daniels CJ. College Algebra. 13th ed. Pearson, 2017.