1994 AMC 12 Problem 23

The L-shaped region has total area $11$

For the desired slope $m,$ the area below the line is the $2$-unit extension plus a triangle of base $3$

The region has area $3\cdot3+2\cdot1=11,$ so each half has area $\frac{11}{2}.$ If the line is $y=mx,$ the part below it consists of the entire $2$-unit rectangle from $x=3$ to $x=5,$ together with a triangle of base $3$ and height $3m.$ Therefore $$2+\frac12(3)(3m)=\frac{11}{2}.$$ This gives $m=\frac{7}{9},$ which indeed lies in the assumed range. Thus the correct answer is E.