#1**0 **

Solve for x over the real numbers:

(66 - 2^x)/(2^x + 3) = (4 - 2^x)/(2^(x + 1) + 6)

Hint: | Expand out terms of the left hand side.

(66 - 2^x)/(2^x + 3) = 66/(2^x + 3) - 2^x/(2^x + 3):

66/(2^x + 3) - 2^x/(2^x + 3) = (4 - 2^x)/(2^(x + 1) + 6)

Hint: | Look for an expression to multiply both sides by in order to clear fractions.

Multiply both sides by 2^(x + 1) + 6:

132 - 2^(x + 1) = 4 - 2^x

Hint: | Move everything to the left hand side.

Subtract 4 - 2^x from both sides:

128 + 2^x - 2^(x + 1) = 0

Hint: | Simplify 128 + 2^x - 2^(x + 1) = 0 by making a substitution.

Simplify and substitute y = 2^x.

128 + 2^x - 2^(x + 1) = 128 - 2^x

= 128 - y:

128 - y = 0

Hint: | Divide both sides by the sign of the leading coefficient of 128 - y.

Multiply both sides by -1:

y - 128 = 0

Hint: | Solve for y.

Add 128 to both sides:

y = 128

Hint: | Perform back substitution on y = 128.

Substitute back for y = 2^x:

2^x = 128

Hint: | Perform the prime factorization of the right hand side.

128 = 2^7:

2^x = 2^7

Hint: | Equate exponents.

Equate exponents of 2 on both sides:

**x = 7**

Guest Sep 17, 2017