Simplify a(x+1)==0 to x+1==0

Question

I feel like I'm missing something very simple. I tried searching for similar answers, because I'm sure I'm not the first to as this question, but I had no luck (perhaps I'm missing the correct search terms).

I would like to simplify an equation of the form, e.g. a(x+1)==0 to x+1==0. Trying FullSimplify[a(x+1)==0] does not work, nor did using Reduce or Refine. Eliminate[(x + 1) == 0, a] just outputs True. How do I get Mathematica to remove terms that multiply a parenthetical expression when it is all equal to zero?

Answer 1

This works

Assuming[a != 0, Simplify[a (x + 1) == 0 ]]

Answer 2

Reduce is also useful in situations like this to decompose the solution space of an equation into its possible cases:

Reduce[a (x + 1) == 0]
(* a == 0 || x == -1 *)

which also gives the alternative solution a==0 in addition to the expected solution x+1==0.

Answer 3

As of 11.3 we have DivideSides:

DivideSides[a (x + 1) == 0, a]

Piecewise[{{1 + x == 0, a != 0}}, a*(1 + x) == 0]

Add an assumption:

DivideSides[a (x + 1) == 0, a, Assumptions -> a != 0]

1 + x == 0

Answer 4

Simplify is not the best instrument in this case. It works fine with this trivial equation (as in the example of Konstantinos above), but may fail in less trivial cases, such as already in the equation a (x+1)==b. Indeed,

Assuming[a != 0, Simplify[a (x + 1) == b]]

(* b == a (1 + x) *)

instead of x+1==b/a.

In this case, (assuming by default that a is nonzero) I use a more stable method dividing the both parts of the equation by a:

Map[Divide[#, a] &, a*(x + 1) == 0]

(*   1 + x == 0  *)

or

Map[Divide[#, a] &, a*(x + 1) == b]

(*  1 + x == b/a  *)

Have fun!