# Inequality Solvers¶

sympy.solvers.inequalities.solve_rational_inequalities(eqs)[source]

Solve a system of rational inequalities with rational coefficients.

Examples

>>> from sympy.abc import x
>>> from sympy import Poly
>>> from sympy.solvers.inequalities import solve_rational_inequalities

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

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

sympy.solvers.inequalities.solve_poly_inequality(poly, rel)[source]

Solve a polynomial inequality with rational coefficients.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x
>>> from sympy.solvers.inequalities import solve_poly_inequality

>>> solve_poly_inequality(Poly(x, x, domain='ZZ'), '==')
[FiniteSet(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'), '==')
[FiniteSet(-1), FiniteSet(1)]

sympy.solvers.inequalities.solve_poly_inequalities(polys)[source]

Solve polynomial inequalities with rational coefficients.

Examples

>>> from sympy.solvers.inequalities import solve_poly_inequalities
>>> from sympy.polys import Poly
>>> 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)[source]

Reduce a system of rational inequalities with rational coefficients.

Examples

>>> from sympy import Poly, 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)[source]

Reduce an inequality with nested absolute values.

Examples

>>> from sympy import Abs, Symbol
>>> from sympy.solvers.inequalities import reduce_abs_inequality
>>> 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)

sympy.solvers.inequalities.reduce_abs_inequalities(exprs, gen)[source]

Reduce a system of inequalities with nested absolute values.

Examples

>>> from sympy import Abs, Symbol
>>> from sympy.abc import x
>>> from sympy.solvers.inequalities import reduce_abs_inequalities
>>> 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)

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

Reduce a system of inequalities with rational coefficients.

Examples

>>> from sympy import sympify as S, Symbol
>>> from sympy.abc import x, y
>>> from sympy.solvers.inequalities 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)[source]

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() doesn’t 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.solvers.inequalities import solve_univariate_inequality
>>> from sympy import Symbol, sin, Interval, S
>>> x = Symbol('x')

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

>>> 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)


sympy.solvers.solveset.solvify