Inequality Solvers

For general cases reduce_inequalities() should be used. Other functions are the subcategories useful for special dedicated operations, and will be called internally as needed by reduce_inequalities.

Note

For a beginner-friendly guide focused on solving inequalities, refer to Reduce One or a System of Inequalities for a Single Variable Algebraically.

Note

Some of the examples below use poly(), which simply transforms an expression into a polynomial; it does not change the mathematical meaning of the expression.

sympy.solvers.inequalities.solve_rational_inequalities(eqs)

Solve a system of rational inequalities with rational coefficients.

Examples

>>> from sympy.abc import x
>>> from sympy import solve_rational_inequalities, Poly

>>> solve_rational_inequalities([[
... ((Poly(-x + 1), Poly(1, x)), '>='),
... ((Poly(-x + 1), Poly(1, x)), '<=')]])
{1}

>>> solve_rational_inequalities([[
... ((Poly(x), Poly(1, x)), '!='),
... ((Poly(-x + 1), Poly(1, x)), '>=')]])
Union(Interval.open(-oo, 0), Interval.Lopen(0, 1))

See also

solve_poly_inequality

sympy.solvers.inequalities.solve_poly_inequality(poly, rel)

Solve a polynomial inequality with rational coefficients.

Examples

>>> from sympy import solve_poly_inequality, Poly
>>> from sympy.abc import x

>>> solve_poly_inequality(Poly(x, x, domain='ZZ'), '==')
[{0}]

>>> solve_poly_inequality(Poly(x**2 - 1, x, domain='ZZ'), '!=')
[Interval.open(-oo, -1), Interval.open(-1, 1), Interval.open(1, oo)]

>>> solve_poly_inequality(Poly(x**2 - 1, x, domain='ZZ'), '==')
[{-1}, {1}]

See also

solve_poly_inequalities

sympy.solvers.inequalities.solve_poly_inequalities(polys)

Solve polynomial inequalities with rational coefficients.

Examples

>>> from sympy import Poly
>>> from sympy.solvers.inequalities import solve_poly_inequalities
>>> from sympy.abc import x
>>> solve_poly_inequalities(((
... Poly(x**2 - 3), ">"), (
... Poly(-x**2 + 1), ">")))
Union(Interval.open(-oo, -sqrt(3)), Interval.open(-1, 1), Interval.open(sqrt(3), oo))

sympy.solvers.inequalities.reduce_rational_inequalities(exprs, gen, relational=True)

Reduce a system of rational inequalities with rational coefficients.

Examples

>>> from sympy import Symbol
>>> from sympy.solvers.inequalities import reduce_rational_inequalities

>>> x = Symbol('x', real=True)

>>> reduce_rational_inequalities([[x**2 <= 0]], x)
Eq(x, 0)

>>> reduce_rational_inequalities([[x + 2 > 0]], x)
-2 < x
>>> reduce_rational_inequalities([[(x + 2, ">")]], x)
-2 < x
>>> reduce_rational_inequalities([[x + 2]], x)
Eq(x, -2)

This function find the non-infinite solution set so if the unknown symbol is declared as extended real rather than real then the result may include finiteness conditions:

>>> y = Symbol('y', extended_real=True)
>>> reduce_rational_inequalities([[y + 2 > 0]], y)
(-2 < y) & (y < oo)

sympy.solvers.inequalities.reduce_abs_inequality(expr, rel, gen)

Reduce an inequality with nested absolute values.

Examples

>>> from sympy import reduce_abs_inequality, Abs, Symbol
>>> x = Symbol('x', real=True)

>>> reduce_abs_inequality(Abs(x - 5) - 3, '<', x)
(2 < x) & (x < 8)

>>> reduce_abs_inequality(Abs(x + 2)*3 - 13, '<', x)
(-19/3 < x) & (x < 7/3)

See also

reduce_abs_inequalities

sympy.solvers.inequalities.reduce_abs_inequalities(exprs, gen)

Reduce a system of inequalities with nested absolute values.

Examples

>>> from sympy import reduce_abs_inequalities, Abs, Symbol
>>> x = Symbol('x', extended_real=True)

>>> reduce_abs_inequalities([(Abs(3*x - 5) - 7, '<'),
... (Abs(x + 25) - 13, '>')], x)
(-2/3 < x) & (x < 4) & (((-oo < x) & (x < -38)) | ((-12 < x) & (x < oo)))

>>> reduce_abs_inequalities([(Abs(x - 4) + Abs(3*x - 5) - 7, '<')], x)
(1/2 < x) & (x < 4)

See also

reduce_abs_inequality

sympy.solvers.inequalities.reduce_inequalities(inequalities, symbols=[])

Reduce a system of inequalities with rational coefficients.

Examples

>>> from sympy.abc import x, y
>>> from sympy import reduce_inequalities

>>> reduce_inequalities(0 <= x + 3, [])
(-3 <= x) & (x < oo)

>>> reduce_inequalities(0 <= x + y*2 - 1, [x])
(x < oo) & (x >= 1 - 2*y)

sympy.solvers.inequalities.solve_univariate_inequality(expr, gen, relational=True, domain=Reals, continuous=False)

Solves a real univariate inequality.

Parameters:

expr : Relational

The target inequality

gen : Symbol

The variable for which the inequality is solved

relational : bool

A Relational type output is expected or not

domain : Set

The domain over which the equation is solved

continuous: bool

True if expr is known to be continuous over the given domain (and so continuous_domain() does not need to be called on it)

Raises:

NotImplementedError

The solution of the inequality cannot be determined due to limitation in sympy.solvers.solveset.solvify().

Notes

Currently, we cannot solve all the inequalities due to limitations in sympy.solvers.solveset.solvify(). Also, the solution returned for trigonometric inequalities are restricted in its periodic interval.

Examples

>>> from sympy import solve_univariate_inequality, Symbol, sin, Interval, S
>>> x = Symbol('x')

>>> solve_univariate_inequality(x**2 >= 4, x)
((2 <= x) & (x < oo)) | ((-oo < x) & (x <= -2))

>>> solve_univariate_inequality(x**2 >= 4, x, relational=False)
Union(Interval(-oo, -2), Interval(2, oo))

>>> domain = Interval(0, S.Infinity)
>>> solve_univariate_inequality(x**2 >= 4, x, False, domain)
Interval(2, oo)

>>> solve_univariate_inequality(sin(x) > 0, x, relational=False)
Interval.open(0, pi)

See also

sympy.solvers.solveset.solvify

solver returning solveset solutions with solve’s output API