What it does
An inequality such as 3x − 4 < 11 is true for a whole range of numbers, not just one. Solving it means finding that range, and this page shows it three ways at once:
- step by step, with the one rule that trips everyone up made obvious: multiply or divide by a negative number and the sign turns round;
- on a number line, with a filled dot when the end number is included (≤ or ≥) and an open dot when it is not (< or >);
- in interval notation, such as (−∞, 2] or [−2, 3), and in words.
It covers the Class 11 maths chapter on linear inequalities: one variable, double inequalities such as −3 ≤ 2x + 1 < 7, and inequalities in two variables, where each one is a half-plane and the solution of a system is the region where the half-planes overlap. The same pictures help in Class 10 when a quadratic’s sign matters.
Use the function gallery to revise intervals as domains and ranges first, then the linear pairs for the lines themselves; the overlapping regions here lead straight into linear programming with the LP solver.
How to use it
- One variable. Type a, b and c in a x + b (sign) c and choose the sign. The number line, the steps and the interval appear at once. The last box lists the whole numbers from −10 to 10 that work.
- Double. A left number and sign appear in front. Both signs must point the same way (−3 ≤ … < 7). The steps do the same thing to all three parts.
- x and y. Up to three rows of a x + b y (sign) c, each with its own colour. Tick a row to use it. Each half-plane is lightly shaded in its colour, and the overlap is darker. Solid boundary lines are included (≤, ≥), dashed ones are not (<, >). Tap the graph to drop a test point; the page says which inequalities it fails. Corner points where boundary lines meet inside the region are listed.
Worked example
The page opens on −2x + 5 ≥ 1. The steps read: take 5 from both sides, −2x ≥ −4; divide both sides by −2, a negative number, so the sign turns round: x ≤ 2. The number line is shaded from 2 off to the left with a filled dot at 2, and the interval is (−∞, 2]: a round bracket at −∞ (infinity is never reached) and a square bracket at 2 (2 is included). The whole numbers that work run from −10 up to 2.
Switch to Double: it fills in −3 ≤ 2x + 1 < 7. Take 1 from all three parts (−4 ≤ 2x < 6), divide by 2 (−2 ≤ x < 3). The interval is [−2, 3) and the whole-number solutions are −2, −1, 0, 1 and 2.
In x and y, the first two rows, x + y ≤ 6 and 2x − y > 0, overlap in a wedge whose corner is at (2, 4), where the lines x + y = 6 and y = 2x cross.
Tips
- To decide which side of a line to shade, test a point not on it, usually (0, 0): if it makes the inequality true, shade its side.
- ∞ is not a number you can reach, so it always gets a round bracket.
- A strict sign (< or >) means the boundary is left out: an open dot on a number line, a dashed line on a graph.
FAQ
Why does dividing by a negative number flip the sign? Because it reverses the order of numbers: 2 < 5, but −2 > −5. Multiplying both sides by −1 mirrors the number line, so “left of” becomes “right of”.
What if a = 0? Then x disappears, and the inequality is either always true (every real x) or never true (no solution). The page says which.
Is anything stored? No. Everything runs in your browser. fibub is not part of NCERT; the topics follow the syllabus, but the words, pictures and examples are our own.
Where the numbers come from
- Solving linear inequalities, interval notation and graphing half-planes are standard school algebra. Wording, pictures and examples are fibub's own; no textbook text, figures or exercises are used.
- Checked against sympy (solve and solve_univariate_inequality) for the one-variable and double inequalities, and against shapely for the area and corners of a shaded region.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.