[tex]AB = \frac{67}{4}\\[/tex]
If [tex]B[/tex] is the midpoint of [tex]\overline{AC}[/tex] then [tex]AB[/tex] is just half of [tex]AC[/tex] and also, [tex]BC[/tex] is just half of [tex]AC[/tex]. This means that [tex]AB[/tex] must be equal to [tex]BC[/tex]. Essentially, [tex]9x +7 = -3x +20[/tex] and we can solve for the of [tex]x[/tex] from that equation.
[tex]9x +7 = -3x +20 \\ 9x +7 +3x = -3x +20 +3x \\ 12x +7 = 20 \\ 12x +7 -7 = 20 -7 \\ 12x = 13 \\ \frac{12x}{12} = \frac{13}{12} \\ x = \frac{13}{12}[/tex]
Now we can solve for [tex]AB[/tex]
[tex]AB = 9(\frac{13}{12}) +7 \\ AB = 3(\frac{13}{4}) +7 \\ AB = \frac{39}{4} +7 \\ AB = \frac{39}{4} +\frac{28}{4} \\ AB = \frac{67}{4}[/tex]
