Half the length LM=(6x-18)/2=3x-9. So the length of MN=3x-9.
But the length of MN is also x-1, therefore x-1=3x-9, 9-1=3x-x, 8=2x, 4=x, that is, x=4.
CHECK
LM is 6x-18=24-18=6 units long. MN is therefore 3 units long (half of LM), which is also x-1=4-1=3.