To solve this inequality, we first factor out the common terms:

(4^x - 16)(2^x - 3) / 2^x - 1 ≤ 0

= (2^(2x) - 2^4)(2^x - 3) / 2^x - 1
= (2^(2x) - 16)(2^x - 3) / 2^x - 1
= 2^x(2^x - 4)(2^x - 3) / 2^x - 1

Now, we will find the critical points by setting the numerator and denominator equal to zero:

2^x = 0
=> x is not defined

2^x - 4 = 0
=> 2^x = 4
=> x = 2

2^x - 3 = 0
=> 2^x = 3
=> x is not defined

Next, we will test the intervals created by the critical points on the inequality. Testing x < 2:

For x < 2,
2^x - 1 > 0
=> (2^x - 4)(2^x - 3) < 0
=> (positive)(negative) < 0
=> This interval does not satisfy the inequality.

Testing 2 < x:

For 2 < x,
2^x - 1 > 0
=> (2^x - 4)(2^x - 3) > 0
=> (positive)(positive) > 0
=> This interval does not satisfy the inequality.

Hence, the solution to the inequality is: x = 2