What is the solution set for this inequality? (x+1)(x-3)(x-7)(x+6)> 0

The zeros of the left side are x ∈ {-6, -1, 3, 7}. The product on the left changes sign at each of these values of x. The product will be positive (relation satisfied) for
  x < -6 . . . . . . even number of negative factors
  -1 < x < 3
  7 < x . . . . . .. no negative factors (0 is also an even number)

The solution set is the union of intervals
  (-∞, -6) ∪ (-1, 3) ∪ (7, ∞)

Since the relation does not include the "or equal to" case, the intervals are all open.

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