When an exam grader marks an inequality problem, they expect to see a handful of specific errors repeat across the class. The syllabus in your head says "I know how to solve inequalities." The exam says "you forgot to flip the sign when you divided by -3, and the entire second half of your answer is wrong." The gap between those two sentences is where most marks are lost.

This satellite is a gallery of five of the most common mistakes — the ones that account for a huge fraction of the red ink in CBSE, ICSE, JEE Main and Advanced inequality questions. Each panel shows a student's worked solution for a particular inequality. The work looks reasonable at first glance. Exactly one step in each solution is wrong. Scrub through the problems with the slider, then press Show the mistake to see the faulty step highlighted, along with the correct number-line shading for comparison.

The widget

Five panels, each with a student's line-by-line solution. The slider picks the problem; the button reveals the one faulty step in red plus the corrected number-line shading below. Use it to train the pattern-recognition: after a while, the bad steps start to jump out on their own.

The five mistakes, one at a time

Mistake 1: forgetting to flip the sign when dividing by a negative

The first panel shows a student solving -3x + 2 > 11. They subtract 2 to get -3x > 9 — correct. Then they divide both sides by -3 and write x > -3. That step is wrong. Dividing (or multiplying) by a negative number reverses the inequality symbol; the correct next line is x < -3.

Why it is wrong. Think of the inequality geometrically. The number line is a ruler with bigger numbers to the right. Multiplying by -1 is a reflection of the line across 0: what was on the right goes to the left, and vice versa. So if a was bigger than b, after reflection -a is smaller than -b. The sign has to flip to keep the statement true. Forgetting this is arithmetic — a single missed symbol — but the answer changes from (-\infty, -3) to (-3, \infty), which is the entirely opposite half of the real line. This is the single most common mistake in inequality problems, and exam graders see it on every paper.

Mistake 2: using an open bracket where the endpoint is included

The second panel solves 5x - 1 \ge 14. The algebra is perfect: x \ge 3. But the student writes the interval as (3, \infty) instead of [3, \infty). The square bracket at 3 would have told the grader that x = 3 is included in the answer; the round bracket says it is not.

Why it is wrong. The symbol \ge means "greater than or equal to" — so equality is allowed. In interval notation this maps directly to a closed (square) bracket and a filled dot on the number line. The open/closed choice is not cosmetic: x = 3 satisfies 5(3) - 1 = 14 \ge 14, so 3 is a solution. Excluding it by writing (3, \infty) drops a valid point from the answer set.

The rule of thumb is simple. Strict inequality (< or >) \rightarrow round bracket, hollow dot. Non-strict (\le or \ge) \rightarrow square bracket, filled dot. Check every endpoint against the original symbol before you write the final answer.

Mistake 3: losing a solution when taking square roots

The third panel tackles x^2 < 9. The student takes the square root of both sides and writes x < 3. That is the kind of answer that looks tidy and feels correct — until you remember that x = -5 also satisfies x < 3 by their logic, but (-5)^2 = 25 > 9, so -5 is not a solution of the original inequality.

Why it is wrong. The square root of x^2 is |x|, not x. Squaring is a two-to-one function: both 3 and -3 square to 9. So when you "undo" the square, you have to account for both roots. The correct move is \sqrt{x^2} < \sqrt{9}, which is |x| < 3, which unpacks to -3 < x < 3 — the interval (-3, 3). The student's answer (-\infty, 3) includes everything below -3, which is wrong, and misses nothing on the right, but it is still wrong because it includes infinitely many non-solutions.

Every time you take a square root in an inequality, write |x| on the first line as a placeholder, then split into the two cases the absolute value produces. That habit stops the mistake before it starts.

Mistake 4: multiplying or dividing by a variable without checking its sign

The fourth panel attempts \frac{1}{x} > 2. The student multiplies both sides by x to get 1 > 2x, then divides by 2 to get x < \frac{1}{2}. In this panel the student assumed x > 0 (stated up front) so the step happens to be safe — but the general lesson is that multiplying or dividing by a variable expression is only safe when you know the sign of that expression.

Why it is wrong in general. If x could be negative, then multiplying both sides by x flips the inequality — the same rule as multiplying by any negative number. Without splitting into cases (x > 0 and x < 0), you risk dropping half of the solution or spuriously including values that do not solve the original. The correct general approach is to rearrange to \frac{1}{x} - 2 > 0, combine to \frac{1 - 2x}{x} > 0, and analyse the sign of the numerator and denominator separately. Doing the sign chart catches both cases at once, and the division-by-variable trap never appears.

Mistake 5: writing a square bracket next to infinity

The fifth panel solves x \le 5 and writes the answer as [-\infty, 5]. The right endpoint is fine — 5 is included, so a square bracket is right. The left side is wrong: \infty (and -\infty) are not real numbers, so you cannot "include" them in a set of real numbers. The bracket next to infinity is always round.

Why it is wrong. A square bracket in interval notation means "this endpoint is an element of the set." The set [-\infty, 5] would claim that -\infty itself is a real number in the solution — but -\infty is not in \mathbb{R}. It is a notational device meaning "there is no lower bound." The convention is universal: (-\infty, 5], (7, \infty), (-\infty, \infty) — round bracket at infinity, every single time.

Meta-lesson: the two checks that catch most errors

Every time you finish an inequality problem, run two checks before you write the final answer.

Check one: sign flip audit. Look back at every multiplication and division you did. If any factor was negative, verify that you flipped the inequality symbol at that step. If the factor was a variable, verify that you either assumed a sign (and said so) or split into cases. This single check catches mistakes 1 and 4.

Check two: bracket audit. Look at every endpoint of your final interval. For each endpoint, ask: does the original inequality symbol (< vs \le, > vs \ge) include this point? If yes, square bracket. If no, round bracket. If the endpoint is \pm\infty, round bracket, no exceptions. This single check catches mistakes 2 and 5.

Mistake 3 — dropping a solution when squaring or square-rooting — is a separate habit: whenever a square or square root appears, write the absolute value explicitly. If you never erase the |\cdot| bars until you have split into cases, you cannot lose half the answer.

Five mistakes, two checks, one habit. Most inequality problems are not hard algebra — they are routine algebra with these five traps laid for you to step into. The students who get full marks are not the ones with better arithmetic; they are the ones who run these checks every time.

A final calibration exercise

Before you close this page, try the widget one more time with the slider set to a random problem — but this time, cover the "Show the mistake" button with your hand and try to find the faulty step on your own first. Pay attention to the steps that involve a sign, a bracket, a square root, or a division. If you can consistently spot the bad step without the reveal, the five patterns have moved from explicit rules in your head to pattern-matching in your eyes, which is the level you need for timed exams.

The goal is not memorising five mistakes. It is recognising the shape of a step that can go wrong — a division by something whose sign is in question, a square that erases the minus sign, a bracket that does not match its symbol. Once you see those shapes, you see the trap before you fall into it.

See also