Here are the facts.
The eastern border of Texas is 94 degrees longitude and the western border of Texas (El Paso) is 106.3 degrees. The difference being 12.3 degrees. If you add 94 degrees to 6.1 degrees (1/5 of 12.3) you get the center of Texas at 100.1 degrees. Abilene is 20 miles east of 100 degrees, therefore, Abilene is technically in the eastern half of the state.
However, the common definition of West Texas is where the grass don't grow, and that's Weatherford.