We need to solve the equation $\frac{x+8}{6} + 3 = \frac{3x+8}{8}$ for $x$.

1. Problem Description

We need to solve the equation $\frac{x+8}{6} + 3 = \frac{3x+8}{8}$ for $x$.

2. Solution Steps

First, we multiply both sides of the equation by the least common multiple of 6 and 8, which is

2

4.

$24(\frac{x+8}{6} + 3) = 24(\frac{3x+8}{8})$

Distribute the 24 on both sides:

$24(\frac{x+8}{6}) + 24(3) = 24(\frac{3x+8}{8})$

Simplify:

$4(x+8) + 72 = 3(3x+8)$

Distribute the 4 and 3:

$4x + 32 + 72 = 9x + 24$

Combine like terms:

$4x + 104 = 9x + 24$

Subtract $4x$ from both sides:

$104 = 5x + 24$

Subtract 24 from both sides:

$80 = 5x$

Divide by 5:

$x = \frac{80}{5}$

$x = 16$

3. Final Answer

$x = 16$