Is there a difference between inequality and equality sign when using Lagrange multiplier?

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For example,

find the extreme values of z=xy subject to the condition x+y=1

This is quite simple example of finding extreme using Lagrange multiplier

When the constrain is changed from x+y=1 to x+y<1 (or less than or equal to ).

Is there a difference when using Lagrange multiplier, or is it just same as equality sign?

Just let inequality be equality and do as in usual way and find every point satisfying

inequality sign?