Question

If \( f(2x) = \frac{2}{2 + x} \) for all \( x>0 \), and \( 5f(x) = 8 \), then what is the value of \( x \)?

• A. \( \frac{-3}{2} \)
• B. \( \frac{-5}{2} \)
• C. \( \frac{3}{2} \)
• D. \( \frac{5}{2} \)

Hint:

For equations involving fractions, always cross-multiply first to eliminate the denominator, then solve for the variable.

Answer 1

The Correct Option is A

We are given that \( 5f(x) = 8 \), so \[ f(x) = \frac{8}{5} \] We also know that \( f(2x) = \frac{2}{2 + x} \). Set \( \frac{2}{2 + x} = \frac{8}{5} \) and solve for \( x \): \[ \frac{2}{2 + x} = \frac{8}{5} \] Cross-multiply: \[ 2 \times 5 = 8 \times (2 + x) \] \[ 10 = 16 + 8x \] \[ 8x = -6 \] \[ x = \frac{-3}{2} \] Thus, the value of \( x \) is \( \frac{-3}{2} \).