Carcass wrote:
If 9^{(x - \frac{1}{2})} – 2^{(2x – 2)} = 4^x – 3^{(2x – 3)}, then what is the value of x ?
A. 3/2
B. 3/4
C. 4/9
D. 2/5
E. 1/5
Given:
9^{(x - \frac{1}{2})} – 2^{(2x – 2)} = 4^x – 3^{(2x – 3)}Rearrange:
9^{(x - \frac{1}{2})} + 3^{(2x – 3)} = 4^x + 2^{(2x – 2)}Rewrite
9 and
4 to get: :
(3^2)^{(x - \frac{1}{2})} + 3^{(2x – 3)} = (2^2)^x + 2^{(2x – 2)}Apply the Power of a Power law to get:
3^{(2x - 1)} + 3^{(2x – 3)} = 2^{2x} + 2^{(2x – 2)}Factor each side as follows:
3^{(2x – 3)}(3^2 + 1) = 2^{(2x – 2)}(2^2 + 1)Evaluate:
3^{(2x – 3)}(10) = 2^{(2x – 2)}(5)Divide both sides by
10 to get:
3^{(2x – 3)} = \frac{2^{(2x – 2)}(5)}{10}Simplify:
3^{(2x – 3)} = \frac{2^{(2x – 2)}}{2}Simplify:
3^{(2x – 3)} = 2^{(2x – 3)}Divide both sides by
2^{(2x – 3)} to get:
\frac{3^{(2x – 3)}}{2^{(2x – 3)}}=1Simplify:
(\frac{3}{2})^{(2x – 3)}=1So, it must be the case that
2x-3 = 0Solve to get:
x = \frac{3}{2}Answer: A
Cheers,
Brent