#1/(2(x-3))+3/(2-x)=5 x#
Make the denominators equal:
#1/(2(x-3)) xx(2-x)/(2-x) +3/(2-x) xx (2(x-3))/(2(x-3))=5 x#
#(2-x)/(2(x-3)(2-x)) + (3xx2(x-3))/(2(x-3)(2-x) =5#
Now we can add the numerators:
#=>( (2-x) + 6(x-3))/(2(x-3)(2-x)) =5#
# (2-x +6x-18)/(2(x-3)(2-x)) =5#
Transposition :
# => (2-x +6x-18) = 5xx2(x-3)(2-x)#
#=> 5x -16 = 10(x(2-x) -3(2-x)#
#=> 5x -16 = 10(2x-x^2 -6+ 3x)#
#=> 5x -16 = 10(5x-x^2 -6)#
#=> 5x -16 = 50x-10x^2 -60#
#=> 50x-5x-10x^2 -60+16 = 0#
#=> -10x^2 +45x -44 =0#
#= 10x^2 -45x+ 44 = 0#
Solve using quadratic formula:
#x=( -b +-sqrt(b^2 -4ac))/(2a)#
Here# a= 10, b= -45 and c= 44#
#b^2 -4ac = 2025-1760 = 265#
We get :
# x = 1.436# or #x = 3.064#