You can multiply both sides of an inequality by the same, nonzero number, but if this number is negative, you have to flip the sign.
In this case, in order to solve for x, you have to divide by -3 (i.e. multiply by -1/3), so yes, you must switch the sign:
[tex]-3x<15 \implies x>-5[/tex]
Alternatively, you may add 3x to both sides:
[tex]0<3x+15[/tex]
Subtract 15 from both sides:
[tex]3x>15[/tex]
And divide both sides by 3:
[tex]x>5[/tex]
The sign switch here was hidden behind the fact that we switched the two sides of the inequality
