Since the vertical line bisects the angle, the segments on the left have the same proportion as those on the right.
.. 8x/(10x -2) = (5x +3)/(7x)
.. 56x^2 = 50x^2 +20x -6 . . . . . "cross multiply"
.. 6x^2 -20x +6 = 0 . . . . . . . . . . subtract the right side
.. 2(3x -1)(x -3) = 0 . . . . . . . . . . . factor
By the zero product rule, one factor must be zero.
.. 3x -1 = 0
.. x = 1/3
.. x -3 = 0
.. x = 3
In order to make the triangle work, we also need to have
(10x -2) > 8x
.. 2x > 2
.. x > 1
and
(7x) > 5x +3
.. 2x > 3
.. x > 1.5
The only value of x satisfying the requirements is x = 3.
