Inequality Solver

An inequality is a mathematical comparison between 2 elements separated by a comparator operator (greater, less, different) and each of the sides that can contain variables/unknowns.

Enter the inequality or inequalities to be solved, and the unknown variables to be found.

Inequalities can be combined, either by writing one inequality per line, either with the and (logical conjunction) operator: && or ⋀.

Solutions are presented with logical simplifications (not in interval notation form).

Inequality solving is similar to equation solving, however, the presence of the comparison sign involves a few additional rules:

— Multiplying by the same strictly negative real number on each side of an inequality changes the direction of the inequality: if $ a < b $ and $ c < 0 $ then $ a \times c > b \times c $

— Multiplying by the same strictly positive real number on each side of an inequality does not change the direction of the inequality: if $ a < b $ and $ c > 0 $ then $ a \times c < b \times c $

— Dividing by the same strictly negative real number on each side of an inequality changes the direction of the inequality: if $ a < b $ and $ c < 0 $ then $ \frac{a}{c} > \frac{b}{c} $

— Dividing by the same strictly positive real number on each side of an inequality does not change the direction of the inequality: if $ a < b $ and $ c > 0 $ then $ \frac{a}{c} < \frac{b}{c} $

— Adding the same real number (positive or negative) on each side of an inequality does not change the direction of the inequality: if $ a < b $ and $ c \in \mathbb{R} $ then $ a + c < b + c $

— Subtracting the same real number (positive or negative) from each side of an inequality does not change the direction of the inequality: if $ a < b $ and $ c \in \mathbb{R} $ then $ a - c < b - c $

— Squaring each positive side of an inequality does not change the direction of the inequality: if $ 0 < a < b $ then $ a^2 < b^2 $

— Squaring each negative side of an inequality changes the direction of the inequality: if $ a < b < 0 $ then $ a^2 > b^2 $

— Inverting each (non-zero) side of an inequality changes the direction of the inequality: if $ a < b $ then $ \frac{1}{a} > \frac{1}{b} $

It is possible to merge several inequalities:

— Add member to member of inequalities of the same direction: if $ a < b $ and $ c < d $ then $ a+c < b+d $

— Multiply member by member of inequalities of the same direction: if $ 0 < a < b $ and $ 0 < c < d $ then $ a \times c < b \times d $

Inequation operators allowed by the calculator are

— < (strictly inferior to, less than)

— <= (less than or equal to)

— > (strictly superior, greater than)

— >= (greater than or equal)

— <> or !=(different, not equal)

The calculating steps from the inequality solver are not displayed because the calculator is based on informatics operations that do not correspond to those of a hand-made resolution.