## STACK Documentation

Documentation home | Category index | Site map# 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.

## See also

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