(e) \( \frac{x+2}{3}-\frac{x+4}{2}=-\frac{1}{6}(x+16) \)

To solve the equation \( \frac{x+2}{3}-\frac{x+4}{2}=-\frac{1}{6}(x+16) \), first, find a common denominator for the fractions on the left-hand side, which is 6. Rewriting the equation gives: \[ \frac{2(x+2)}{6} - \frac{3(x+4)}{6} = -\frac{1}{6}(x+16). \] This simplifies to: \[ \frac{2(x+2) - 3(x+4)}{6} = -\frac{1}{6}(x+16). \] Now, clear the fractions by multiplying the entire equation by 6: \[ 2(x+2) - 3(x+4) = - (x + 16). \] Expanding both sides results in: \[ 2x + 4 - 3x - 12 = -x - 16. \] Combining like terms gives: \[ -x - 8 = -x - 16. \] Adding \( x \) to both sides cancels the \( -x \): \[ -8 = -16, \] which is **not true**. This indicates that there is no solution; the equation is inconsistent. Now, here's a tip! Sometimes, in linear equations, you can end up with a statement that doesn't hold true, just like this one. This usually means the lines represented by the equation don’t intersect. So, always check your steps for common mistakes that can lead to no solution. Happy solving!