Built-in Mathematica Symbol

LessEqual (<=, ≤)

x<=y or x≤y
yields True if

is determined to be less than or equal to

.

x₁≤x₂≤x₃
yields True if the

form a nondecreasing sequence.

MORE INFORMATION

can be entered as Esc <= Esc 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 ≤.

, entered as Esc </ Esc or x\[LessSlantEqual]y, can be used on input as an alternative to .

EXAMPLES

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Basic Examples (2)

Compare numbers:

Represent an inequality:

Scope (9)

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 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 ten digits:

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

The extra digits in this case are ignored by LessEqual: