# Inequalities

The non-strict inequalities $\geq$ and $\leq$ are created as infix operators with the respective syntax

>=,  <=


Maxima allows single inequalities, such as $x-1>y$, and also support for inequalities connected by logical operators, e.g. $x>1 \mbox{ and } x<=5$.

You can test if two inequalities are the same using the algebraic equivalence test, see the comments on this below.

Chained inequalities, for example $1\leq x \leq2\mbox{,}$ are not permitted. They must be joined by logical connectives, e.g. "$x>1$ and $x<7$".

From version 3.6, support for inequalities in Maxima (particularly single variable real inequalities) was substantially improved.

# Functions to support inequalities

• ineqprepare(ex)

This function ensures an inequality is written in the form ex>0 or ex>=0 where ex is always simplified. This is designed for use with the algebraic equivalence answer test in mind.

• ineqorder(ex)

This function takes an expression, applies ineqprepare(), and then orders the parts. For example,

 ineqorder(x>1 and x<5);


returns

  5-x > 0 and x-1 > 0


It also removes duplicate inequalities. Operating at this syntactic level will enable a relatively strict form of equivalence to be established, simply manipulating the form of the inequalities. It will respect commutativity and associativity and and and or, and will also apply not to chains of inequalities.

If the algebraic equivalence test detects inequalities, or systems of inequalities, then this function is automatically applied.

• make_less_ineq(ex)

Reverses the order of any inequalities so that we have A<B or A<=B. It does no other transformations. This is useful because when testing equality up to commutativity and associativity we don't perform this transformation. We need to put all inequalities a particular way around. See the EqualComAss test examples for usage.