Any number inside the modulus sign becomes positive. This means [tex] |15+m|=|-(15+m)|=|-15-m| [/tex] and so we have,
[tex] -2m+3=|15+m| \Rightarrow \left \Big\{ {{-2m+3=15+m} \atop{-2m+3=-15-m}} \right[/tex]
Solving these gives us
[tex] -2m+3=15+m \Rightarrow m=-4 [/tex]
[tex] -2m+3=-15-m \Rightarrow = m=18 [/tex]
However if we check the second solution in the original equation we obtain [tex] -2(18)+3=|15+18| \Rightarrow -33=33 [/tex]. This is false and so [tex] m=18 [/tex] can't be a solution.
Therefore the only solution is [tex] m=-4 [/tex].
(Note: I'm not sure why the second solution didn't work but when there's a modulus sign involved it always pays to check your final answers to be sure. I'll have a think about it but in case you find out before I do, I'd be interested to know in the comments.)
