# How to force simplify to drop common denominator in inequalities?

Simplify[c>0 && a/c <= b/c] results in c > 0 && a <= b, as hoped. However when a and b are even slightly complicated expressions, rather than variables, then even if c is still a variable, then Simplify no longer drops c. Is there a way to force this simplification to be performed?

• Can you also give a (simple) example of a case where your desired simplification does not happen? The behavior may depend on the structure of the expression you are simplifying. Mar 18, 2019 at 16:18
• As an example, c>0 && a/c <= (Sqrt[e^2])/c is simplified to c > 0 && (a - Sqrt[e^2])/c <= 0, while c>0 && (a*d)/c <= (b*Sqrt[e^2])/c is not simplified at all. Mar 19, 2019 at 19:33

With version 11.3 use MultiplySides

ineq = a/c <= b/c;


If you know that the denominator is positive,

ineq2 = Assuming[c > 0, MultiplySides[ineq, c]]

(* a <= b *)


Or, if you know the denominator is negative,

ineq3 = Assuming[c < 0, MultiplySides[ineq, c]]

(* b <= a *)


EDIT: Also, look at Reduce

Reduce[c > 0 && a/c <= b/c]

(* b ∈ Reals && a <= b && c > 0 *)

Assuming[c > 0 && Element[b, Reals],
Reduce[c > 0 && a/c <= b/c] // Simplify]

(* a <= b *)

• Unfortunately, what I am trying to do is not just to simplify a single inequality, but rather simplify a complex boolean combination of a bunch of inequalities... Mar 19, 2019 at 19:26
• That said, Simplify[MultiplySides[c>0 && (a*d)/c <= (b*Sqrt[e^2])/c, c] does produce c > 0 && b*Sqrt[e^2] >= a*d, so it might be workable - I will try it in my actual example. Mar 19, 2019 at 19:37

You should provide the c>0 part as an assumption to the Simplify function. Try e.g.

Simplify[(a*d)/c <= (b*Sqrt[e^2])/c, c > 0]
Assuming[c > 0, Simplify[(a*d)/c <= (b*Sqrt[e^2])/c]]

(* Out: a*d <= b*Sqrt[e^2] *)


Both remove c` from the denominator.

• This works with slightly more complex inequalities, but not by far. Also, it does not help simplify more Boolean combinations of inequality, where c>0 is part of a local clause, but is not true for other clauses... Mar 20, 2019 at 20:26