Inequality Calculator Solver
Enter two values and choose a comparison operator (<, ≤, >, ≥) to solve inequalities and see the solution set in interval notation.
Types of Inequality Symbols
Four main symbols express inequality relationships. The less-than symbol < means the left side is smaller than the right. The greater-than symbol > means the left side is larger. These are strict inequalities because the two sides cannot be equal.
Adding an underline creates inclusive inequalities. The symbol ≤ means "less than or equal to," while ≥ means "greater than or equal to." These allow the two sides to match exactly, expanding the solution set by one point—the boundary value itself.
Understanding which symbol to use depends on context. Speed limits typically use ≤ because you can legally drive at exactly the limit. Temperature warnings might use > if a threshold triggers an alert only when exceeded, not when matched.
Solving Linear Inequalities
Solving linear inequalities follows the same steps as solving linear equations: simplify both sides, isolate the variable, and perform inverse operations. The critical rule that changes everything is the sign flip rule—whenever you multiply or divide both sides by a negative number, reverse the inequality direction.
For example, solving -3x < 9 requires dividing by -3. The division flips the inequality to x > -3. This rule exists because multiplying by negative numbers reverses order on the number line. Since -2 < -1, multiplying both sides by -1 gives 2 > 1, which is true.
After solving, always express your answer in interval notation or graph it. This clarifies the solution set and makes it easier to check your work by testing a value from the range.
Compound Inequalities and Interval Notation
Compound inequalities combine two inequality statements using "and" or "or." The statement 1 < x < 5 is shorthand for x > 1 AND x < 5, describing all numbers between 1 and 5. An "or" compound like x < 0 or x > 10 includes two separate ranges.
Interval notation streamlines writing these solutions. The compound 1 < x < 5 becomes (1, 5). Parentheses mean open endpoints—the boundary values are not included. Brackets mean closed endpoints. So 1 ≤ x ≤ 5 becomes [1, 5]. Mixed endpoints like 1 < x ≤ 5 give (1, 5].
"Or" compounds use the union symbol ∪. The solution x < 0 or x > 10 becomes (-∞, 0) ∪ (10, ∞). Infinity always gets a parenthesis because you can't reach or include infinity as a value.
Frequently Asked Questions
What is an inequality?
An inequality is a mathematical statement comparing two expressions using <, ≤, >, or ≥. Unlike equations, inequalities describe a range of solutions rather than a single value.
How do you solve an inequality?
Solve inequalities like equations by isolating the variable. The key difference: when you multiply or divide by a negative number, flip the inequality sign. For example, solving -2x > 6 gives x < -3.
What is interval notation?
Interval notation expresses solution sets using brackets. Parentheses ( ) mean the endpoint is not included, brackets [ ] mean it is. For example, (2, 5] means x > 2 and x ≤ 5.
What does the empty set symbol mean?
The symbol ∅ represents the empty set, meaning no values satisfy the inequality. This happens when the statement is always false, like 5 < 3.
How are inequalities used in real life?
Inequalities model constraints in real situations. Budget limits (cost ≤ $500), speed limits (speed < 65 mph), and temperature ranges (32°F ≤ temp ≤212°F) all use inequality notation.