How did you do that, I really don't see where the -10, 13 and 15 came from unless im being really stupid :|
You seem to have kept the loosest bounds and discarded the strictest.
$\displaystyle \displaystyle x \ge 0$
$\displaystyle 7-x \ge 0 \implies x \le 7$
$\displaystyle 6-x \ge 0 \implies x \le 6$
$\displaystyle 4+x \ge 0 \implies x \ge -4$
Discarding the second and fourth, I get $\displaystyle 0 \le x \le 6$ .
IMPORTANT: x is not the value you seek to find the min and max of. You want to find min and max of "the number of male patients under 50 without a back problem" which is 7-x. So...