# LessEqual x<=y or xy

yields True if x is determined to be less than or equal to y.

x1x2x3

yields True if the form a nondecreasing sequence.

# Details • LessEqual is also known as weak inequality or non-strict inequality.
• xy can be entered as x <= y or x \[LessEqual]y .
• LessEqual gives True or False when its arguments are real numbers.
• LessEqual does some simplification when its arguments are not numbers.
• For exact numeric quantities, LessEqual internally uses numerical approximations to establish numerical ordering. This process can be affected by the setting of the global variable \$MaxExtraPrecision.
• In StandardForm, LessEqual is printed using .
• xy, entered as x </ y or x \[LessSlantEqual] y, can be used on input as an alternative to xy.

# Examples

open allclose all

## Basic Examples(2)

Compare numbers:

Represent an inequality:

## Scope(9)

### Numeric Inequalities(7)

Inequalities are defined only for real numbers: Compare rational numbers:

Approximate numbers that differ in at most their last eight binary digits are considered equal:

Compare an exact numeric expression and an approximate number:

Compare two exact numeric expressions; a numeric test may suffice to prove inequality:

Proving this inequality requires symbolic methods:

Symbolic and numeric methods used by LessEqual are insufficient to prove this inequality: Use RootReduce to decide the sign of algebraic numbers:

Numeric methods used by LessEqual do not use sufficient precision to disprove this inequality: RootReduce disproves the inequality using exact methods:

Increasing \$MaxExtraPrecision may also disprove the inequality:

### Symbolic Inequalities(2)

Symbolic inequalities remain unevaluated, since x may not be a real number:

Use Refine to reevaluate the inequality assuming that x is real:

A symbolic inequality:

Use Reduce to find an explicit description of the solution set:

Use FindInstance to find a solution instance:

Use Minimize to optimize over the inequality-defined region:

Use Refine to simplify under the inequality-defined assumptions:

## Properties & Relations(12)

The negation of two-argument LessEqual is Greater:

The negation of three-argument LessEqual does not simplify automatically:

Use LogicalExpand to express it in terms of two-argument Greater:

This is not equivalent to three-argument Greater:

When LessEqual cannot decide inequality between numeric expressions, it returns unchanged: FullSimplify uses exact symbolic transformations to prove the inequality:

NonPositive[x] is equivalent to :

Use Reduce to solve inequalities:

Use FindInstance to find solution instances:

Use RegionPlot and RegionPlot3D to visualize solution sets of inequalities:

Inequality assumptions:

Use Minimize and Maximize to solve optimization problems constrained by inequalities:

Use NMinimize and NMaximize to numerically solve constrained optimization problems:

Integrate a function over the solution set of inequalities:

Use Median, Quantile, and Quartiles to the  greatest number(s):

## Possible Issues(3)

Inequalities for machine-precision approximate numbers can be subtle:

The result is based on extra digits:

Arbitrary-precision approximate numbers do not have this problem:

Thanks to automatic precision tracking, LessEqual knows to look only at the first 10 digits:

In this case, inequality between machine numbers gives the expected result:

The extra digits in this case are ignored by LessEqual: