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Rational Inequalities Calculator

SOLUTION SET
—
Choose the sign, fill in both sides, then press =.
The sign across the number line
top
bottom
top
bottom
You cannot multiply both sides by the bottom — its sign is unknown, and a negative one would flip the inequality. Everything goes to one side instead and the sign is read off a number line.
How this inequality is solved, step by step
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The first thing most people do with 1/(x−2) > 0 is multiply both sides by x−2. That is the one move you are not allowed to make.

You do not know the sign of x−2. If it is negative, multiplying by it turns the > into a <, and you cannot tell which case you are in until the thing is already solved. This rational inequalities calculator never multiplies up. It moves everything to one side and reads the sign off a number line instead, which is the method that actually works.

Rational Inequalities Calculator logo — a free tool from Monkza

How It Solves It, with Steps

Everything goes to one side and over a single denominator. The values that make the top zero and the values that make the bottom zero become the critical points, and between two consecutive ones the expression cannot change sign — so a single test value decides the whole interval. The number line shows every interval with its sign, the ones in the answer marked in green.

Then each critical point is labelled. A zero of the top is included when the sign is ≥ or ≤, because the expression equals zero there. A zero of the bottom is never allowed, whatever sign you chose, because the expression is undefined there — and that is the mark most often lost on these questions. Solving rational inequalities comes down to those two rules more than to any algebra.

The answer comes out in interval notation, square brackets where the endpoint is in and round ones where it is out. Fractional critical points stay exact, never decimals.

How to Use This Rational Inequalities Calculator

Pick one of the four signs, fill in the top and bottom of each side, and press =. The right side can be a plain number — write it over 1 — or a second fraction.

The solution set appears with the values x can never take beneath it. One tap opens the number line and the point table, and under that the working step by step. Free to use online, with no account.

Frequently Asked Questions

Why can't you multiply both sides by the denominator?
Because you do not know its sign. If the denominator turns out to be negative, multiplying by it reverses the inequality, and you have no way of knowing which case you are in until you have already solved it. Moving everything to one side avoids the question.
How do you solve a rational inequality?
Get everything onto one side over a single denominator, find the values that make the top or the bottom zero, and test one number in each interval those values create. The sign cannot change inside an interval, so one test settles the whole stretch.

Never multiply up, take the critical points from the top and the bottom both, and remember that a bottom zero gets a round bracket even with ≥.