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In order for g(x) be positive, both the numerator and denominator should be EITHER both positive, OR both negative.
So, we consider two cases.
Case (a). x+8 > 0 AND x+2 > 0.
The solution set to these two inequalities is {x > - 8} AND {x > - 2}.
The last two inequalities are EQUIVALENT to one single inequality x > -2.
Case (b). x+8 < 0 AND x+2 < 0.
The solution set to these two inequalities is {x < - 8} AND {x < - 2}.
The last two inequalities are EQUIVALENT to one single inequality x < -8.
Thus the final solution to your problem is this:
{x < -8} or {x > -2},
Or in the interval form
(-oo,-8) U {-2,oo) (union of two sets).
Solved.
