In triangles with the same altitude, area is proportional to the base length. Triangles CDE and BDE both have the same altitude (the perpendicular distance from D to BC), so their areas are in the proportion BE:EC = 1:5. Since ACDE = 5 in², ABDE = 1 in². The area of BDC is the total of those:
... ABDC = 6 in²
Likewise, triangles ABD and CBD both have an altitude that is the perpendicular distance from B to AC. Then, ...
... AABD : ACBD = AD : CD = 4:3
Thus AABD = 4/3 × ACBD = 4/3 × 6 in²
... AABD = 8 in²
Of course, AABC is the sum of AABD and ABDC, so is ...
... AABC = 6 in² + 8 in²
... AABC = 14 in²