# Strange behavior when solving absolute value equations

Try this:

Solve[Abs[x/2 - 1/3] == 1, x]


Produces this answer:

(* {{x -> -(4/3)}, {x -> 8/3}} *)


Now try this:

Solve[Abs[2 x - 3] == 7, x]


Which gives this error message and answer:

Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. >>

(* {{x -> -2}, {x -> 5}} *)


Now I know you can cure the problem like this:

Solve[Abs[2 x - 3] == 7, x, Reals]


But doesn't it seem strange that they are quite similar and one works and one doesn't?

• A little experimentation suggests that it's the == 1 that is making it work without spitting out an error: try Solve[Abs[2 x - 3] == 1, x]. I can't guess as to why this is. – march Jan 23 '16 at 5:22
• Indeed strange, notice in the first case Solve doesn't find all the solutions, too. Just try Reduce[Abs[x/2 - 1/3] == 1, x]. – xzczd Jan 23 '16 at 7:11