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# LessEqual

 or yields True if x is determined to be less than or equal to y. yields True if the form a nondecreasing sequence.
• can be entered as x Esc <= Esc y or .
• 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.
• , entered as x Esc </ Esc y or x\[LessSlantEqual]y, can be used on input as an alternative to .
Compare numbers:
Represent an inequality:
Compare numbers:
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Represent an inequality:
 Out[1]=
 Out[2]=
 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:
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):
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:
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