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 - = (2^(2x) - 16)(2^x - 3) / 2^x - = 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 = => x is not defined
2^x - 4 = => 2^x = => x = 2
2^x - 3 = => 2^x = => x is not defined
Next, we will test the intervals created by the critical points on the inequality. Testing x < 2:
For x < 22^x - 1 > => (2^x - 4)(2^x - 3) < => (positive)(negative) < => This interval does not satisfy the inequality.
Testing 2 < x:
For 2 < x2^x - 1 > => (2^x - 4)(2^x - 3) > => (positive)(positive) > => This interval does not satisfy the inequality.
Hence, the solution to the inequality is: x = 2
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 -
= (2^(2x) - 16)(2^x - 3) / 2^x -
= 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 =
=> x is not defined
2^x - 4 =
=> 2^x =
=> x = 2
2^x - 3 =
=> 2^x =
=> 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 >
=> (2^x - 4)(2^x - 3) <
=> (positive)(negative) <
=> This interval does not satisfy the inequality.
Testing 2 < x:
For 2 < x
2^x - 1 >
=> (2^x - 4)(2^x - 3) >
=> (positive)(positive) >
=> This interval does not satisfy the inequality.
Hence, the solution to the inequality is: x = 2