x^2 -2x-8<7|x-4|

--------[4]-------------

1)x<4       x^2-2x-8<-7x+28

                 x^2+5x-36<0

                (x+9)(x-4)<0

                     +                 -                 +

                ---------(-9)-----------(4)---------------

                                (-9;4)

2)x>=4      x^2-2x-8<7x-28

                x^2-9x+20<0

                (x-5)(x-4)<0

                        +               -                  +

                -------------(4)----------(5)----------

                       (4;5)

   х принадлежит (-9;4)объединено  (4;5)

Verified answer

1) x>=4, то

x^2-2x-8<7x-28

x^2-9x+20<0

x1=4,  x2=5

4<x<5

2) x<4, то 

x^2-2x-8<-7x+28

x^2+5x-36<0

x1=-9,   x2=4

-9<x<4

Ответ :  х принадлежит  (-9; 4) U(4;  5)