x<=y or x≤y
yields True if x is determined to be less than or equal to y.
yields True if the form a nondecreasing sequence.
- LessEqual is also known as weak inequality or non-strict inequality.
- x≤y 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 ≤.
- x≤y, entered as x </ y or x \[LessSlantEqual] y, can be used on input as an alternative to x≤y.
Examplesopen allclose all
Numeric Inequalities (7)
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)
Use Refine to reevaluate the inequality assuming that x is real:
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 three-argument LessEqual does not simplify automatically:
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:
Integrate a function over the solution set of inequalities:
Possible Issues (3)
Thanks to automatic precision tracking, LessEqual knows to look only at the first 10 digits:
The extra digits in this case are ignored by LessEqual:
Wolfram Research (1988), LessEqual, Wolfram Language function, https://reference.wolfram.com/language/ref/LessEqual.html (updated 1996).
Wolfram Language. 1988. "LessEqual." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 1996. https://reference.wolfram.com/language/ref/LessEqual.html.
Wolfram Language. (1988). LessEqual. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LessEqual.html