To solve the inequality 16/(x^2-6x-7)≤0, we first need to find the critical points.

To do this, we first factor the denominator x^2-6x-7 using the quadratic formula:

x = (6 ± √(6^2 - 4(1)(-7)))/(2(1))
x = (6 ± √(36 + 28))/2
x = (6 ± √64)/2
x = (6 ± 8)/2
x = 7 or x = -1

Therefore, the critical points are x = 7 and x = -1.

We can now create a number line to test the intervals created by these critical points:

Test x = -2: 16/((-2)^2 - 6(-2) - 7) = 16/(4 + 12 - 7) = 16/9 > 0
Test x = 0: 16/(0^2 - 6(0) - 7) = 16/(-7) < 0
Test x = 5: 16/(5^2 - 6(5) - 7) = 16/(25-30-7) = 16/(-12) < 0
Test x = 8: 16/(8^2 - 6(8) - 7) = 16/(64-48-7) = 16/9 > 0

From our tests, we see that the inequality is satisfied for x in the intervals (-∞, -1) and (7, ∞). Therefore, the solution to the inequality is:

x ∈ (-∞, -1) U (7, ∞)