x+7-8x/3=17/6-5x/2

Add $5 \frac{x}{2}$ to both sides

$x+7-8 \cdot \frac{x}{3}+5 \cdot \frac{x}{2}=\frac{17}{6}-5 \cdot \frac{x}{2}+5 \cdot \frac{x}{2}$

Simplify

$\frac{5 x}{6}+7=\frac{17}{6}$

Subtract 7 from both sides

$\frac{5 x}{6}+7-7=\frac{17}{6}-7$

Simplify

$\frac{5 x}{6}=-\frac{25}{6}$

Multiply both sides by 6

$\frac{6 \cdot 5 x}{6}=6\left(-\frac{25}{6}\right)$

Simplify

$5 x=-25$

Divide both sides by 5

$\frac{5 x}{5}=\frac{-25}{5}$

Simplify

$x=-5$