Inequalities and systems of inequalities

The app solves quadratic and rational inequalities with the zero points and a sign chart. A double inequality and a system of inequalities are solved as the intersection of the solutions of the individual inequalities, shown on a number line. A system of inequalities in two unknowns is solved as a region in the plane, shaded in the graph.

In a system of inequalities in two unknowns the app also lists the region’s corners – the intersections of the boundaries with exact coordinates. A system of an equation and inequalities in one unknown is solved by solving the equation and substituting its roots into the inequalities; in two unknowns the solution is the part of the equation’s curve lying in the inequalities’ region – highlighted in the graph, its end points computed exactly. A system can also be written on one line, separated by semicolons (2ˣ > 4; 3ˣ < 81).

Examples

A quadratic inequality – zero points and signs.
Input: x^2 - x - 6 > 0
x^2 - x - 6 > 0
x^2 - x - 6 > 0x^2 - x - 6 > 0
A rational inequality – the denominator must not be zero.
Input: (x+1)/(x-3) <= 0
(x+1)/(x-3) <= 0
(x+1)/(x-3) <= 0(x+1)/(x-3) <= 0
A double inequality.
Input: -1 <= 2x + 1 < 5
-1 <= 2x + 1 < 5
-1 <= 2x + 1 < 5-1 <= 2x + 1 < 5
A system of inequalities in two unknowns – a region in the plane and its corners.
Input: x + y <= 4
x - y >= 0
y >= 0
x + y <= 4 x - y >= 0 y >= 0
x + y <= 4 x - y >= 0 y >= 0x + y <= 4 x - y >= 0 y >= 0
An inequality in the integers – the integers of the interval are listed.
Input: x^2 < 10
x^2 < 10
x^2 < 10x^2 < 10
An equation and an inequality – the equation’s roots substituted into the inequality.
Input: x^2 - x - 6 = 0
x > 0
x^2 - x - 6 = 0 x > 0
A circle and a half-plane – the solution is an arc, its end points don’t belong to it.
Input: x^2 + y^2 = 4
y > x
x^2 + y^2 = 4 y > x
x^2 + y^2 = 4 y > xx^2 + y^2 = 4 y > x

Known limitations