Analyze inequality steps, interval notation, and boundary rules. Review tables, plots, and downloadable results quickly. Learn solution sets with clear examples and guidance today.
| Inequality | Solved Form | Interval Notation |
|---|---|---|
| 2x + 3 < 11 | x < 4 | (-∞, 4) |
| -3x + 6 ≥ 0 | x ≤ 2 | (-∞, 2] |
| 1 ≤ 2x - 1 < 7 | 1 ≤ x < 4 | [1, 4) |
| 5x - 10 > 15 | x > 5 | (5, ∞) |
| 4 ≤ x + 2 ≤ 8 | 2 ≤ x ≤ 6 | [2, 6] |
For a linear inequality of the form ax + b ? c, isolate x using the same algebra steps used for equations.
Subtract b from both sides first.
Then divide both sides by a.
If a is negative, reverse the inequality sign after division.
General form:
ax + b ? c
ax ? c - b
x ? (c - b) / a
When solving a compound inequality, solve both sides separately and intersect the two solution sets.
Interval notation uses parentheses for excluded endpoints and brackets for included endpoints.
Solving inequalities is a core skill in algebra, pre calculus, and many applied maths topics. Unlike equations, inequalities describe ranges of values rather than a single answer. That makes interval notation useful because it compresses the final solution into a compact, standard format that teachers, students, and software tools all recognize.
A typical linear inequality starts with an expression such as ax + b on one side and a constant or another simple value on the other. The goal is to isolate x using algebraic operations. You can add, subtract, multiply, or divide both sides, but one rule matters most: if you divide or multiply by a negative number, the inequality direction must reverse. Many mistakes happen at this step.
Interval notation then converts the solution into endpoint based form. Parentheses mean the endpoint is not included. Brackets mean the endpoint is included. For example, x < 4 becomes (-∞, 4), while x ≤ 4 becomes (-∞, 4]. A bounded solution such as 2 ≤ x < 7 becomes [2, 7). If no values satisfy the inequality, the solution is the empty set, written as ∅.
Compound inequalities combine two comparisons at once. These often appear in the form a < bx + c ≤ d. To solve them, isolate the variable across all parts or solve each side separately and intersect the results. The final answer often becomes a closed, open, or half open interval.
This calculator helps reduce sign errors, shows interval notation clearly, and gives a quick graph view of the solution set. It is useful for homework checking, classroom demonstrations, and practice with inequality transformations.
Interval notation is a compact way to show all values that satisfy an inequality. It uses parentheses for excluded endpoints and brackets for included endpoints.
You reverse the sign only when multiplying or dividing both sides by a negative number. This keeps the comparison mathematically correct.
A bracket means the endpoint is included in the solution set. It matches ≤ or ≥ in the solved inequality.
A parenthesis means the endpoint is not included. It matches < or > in the solved inequality.
Yes. Use compound mode to enter a lower bound, an expression, and an upper bound. The calculator solves both sides and intersects them.
The result is the empty set, written as ∅. This means the inequality has no valid solution values.
It means every real number satisfies the inequality. In interval notation, this appears as (-∞, ∞).
Graphing helps you see endpoint inclusion and the direction of the solution set. It also makes interval answers easier to verify quickly.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.