Inequality Solver
Solve linear, compound, quadratic, rational, and absolute value inequalities with interval notation, number line graphs, step-by-step work, and clear student-friendly explanations.
Background
An inequality compares expressions using symbols like <, ≤, >, and ≥. Solving an inequality means finding every value that makes the statement true. Unlike equations, inequalities often describe a range of answers, which can be written using inequality notation, interval notation, or a number line.
How to use this inequality solver
- Type an inequality such as 2x + 5 ≤ 11.
- Choose Auto-detect or select the type of inequality you want to solve.
- Click Solve to see the answer in inequality notation and interval notation.
- Use the number line to understand which values are included in the solution set.
- Read the step-by-step explanation to see how the inequality was solved.
How this calculator works
- For linear inequalities, it isolates the variable using inverse operations.
- If both sides are divided or multiplied by a negative number, the inequality symbol flips.
- For compound inequalities, it solves each boundary and combines the solution using AND or OR logic.
- For quadratic and rational inequalities, it identifies critical values and tests intervals on the number line.
- For absolute value inequalities, it rewrites the problem as a compound inequality.
Formula & Rules Used
Linear inequality form: ax + b < c, ax + b ≤ c, ax + b > c, or ax + b ≥ c
Sign-flip rule: When multiplying or dividing both sides by a negative number, reverse the inequality symbol.
Interval notation: Parentheses ( ) mean an endpoint is not included, while brackets [ ] mean an endpoint is included.
Compound AND inequality: The solution must satisfy both inequalities.
Compound OR inequality: The solution can satisfy either inequality.
Example Problem & Step-by-Step Solution
Example 1 — Solve a linear inequality
- Start with 2x + 5 ≤ 11.
- Subtract 5 from both sides: 2x ≤ 6.
- Divide both sides by 2: x ≤ 3.
- The solution in interval notation is (−∞, 3].
- On the number line, use a closed circle at 3 and shade to the left.
Example 2 — Watch for the sign flip
- Start with −3x + 4 > 10.
- Subtract 4 from both sides: −3x > 6.
- Divide both sides by −3.
- Because you divided by a negative number, flip the symbol: x < −2.
- The solution in interval notation is (−∞, −2).
Example 3 — Solve a compound inequality
- Start with −3 < 2x + 1 ≤ 7.
- Subtract 1 from all three parts: −4 < 2x ≤ 6.
- Divide all three parts by 2: −2 < x ≤ 3.
- The interval notation is (−2, 3].
- On the number line, use an open circle at −2, a closed circle at 3, and shade between them.
Common mistakes with inequalities
- Forgetting to flip the inequality symbol after dividing by a negative number.
- Using brackets in interval notation when an endpoint should not be included.
- Confusing AND inequalities with OR inequalities.
- Including values that make a denominator equal to zero in rational inequalities.
- Solving absolute value inequalities as if they were ordinary linear inequalities.
FAQs
What is an inequality solver?
An inequality solver is a tool that finds all values of a variable that make an inequality true. Instead of giving one single answer like many equations do, inequalities often give a range of possible values.
When do you flip the inequality sign?
You flip the inequality sign when you multiply or divide both sides of an inequality by a negative number. For example, dividing by −3 changes > to <.
What does interval notation mean?
Interval notation is a compact way to show a solution range. Parentheses mean an endpoint is not included, while brackets mean an endpoint is included.
What is the difference between AND and OR inequalities?
An AND inequality means a value must satisfy both conditions at the same time. An OR inequality means a value can satisfy either condition.
Can this calculator show the answer on a number line?
Yes. The calculator includes a number line graph so students can see whether endpoints are open or closed and which direction or interval should be shaded.
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