a: \(\dfrac{x+4}{3}+\dfrac{x+3}{4}=\dfrac{x+2}{5}+\dfrac{x+1}{6}\)
=>\(\dfrac{x+4}{3}+1+\dfrac{x+3}{4}+1=\dfrac{x+2}{5}+1+\dfrac{x+1}{6}=1\)
=>\(\dfrac{x+7}{3}+\dfrac{x+7}{4}=\dfrac{x+7}{5}+\dfrac{x+7}{6}\)
=>\(\left(x+7\right)\left(\dfrac{1}{3}+\dfrac{1}{4}-\dfrac{1}{5}-\dfrac{1}{6}\right)=0\)
=>x+7=0
=>x=-7
b: \(\dfrac{x+1}{87}+\dfrac{x+3}{85}=\dfrac{x+5}{83}+\dfrac{x+7}{81}\)
=>\(\left(\dfrac{x+1}{87}+1\right)+\left(\dfrac{x+3}{85}+1\right)=\left(\dfrac{x+5}{83}+1\right)+\left(\dfrac{x+7}{81}+1\right)\)
=>\(\dfrac{x+88}{87}+\dfrac{x+88}{85}=\dfrac{x+88}{83}+\dfrac{x+88}{81}\)
=>\(\left(x+88\right)\left(\dfrac{1}{87}+\dfrac{1}{85}-\dfrac{1}{83}-\dfrac{1}{81}\right)=0\)
=>x+88=0
=>x=-88
c: \(\dfrac{x-29}{31}-\dfrac{x-27}{33}=\dfrac{x-17}{43}-\dfrac{x-15}{45}\)
=>\(\dfrac{x-29}{31}+\dfrac{x-15}{45}=\dfrac{x-17}{43}+\dfrac{x-27}{33}\)
=>\(\left(\dfrac{x-29}{31}-1\right)+\left(\dfrac{x-15}{45}-1\right)=\left(\dfrac{x-17}{43}-1\right)+\left(\dfrac{x-27}{33}-1\right)\)
=>\(\dfrac{x-60}{31}+\dfrac{x-60}{45}=\dfrac{x-60}{43}+\dfrac{x-60}{33}\)
=>\(\left(x-60\right)\left(\dfrac{1}{31}+\dfrac{1}{45}-\dfrac{1}{43}-\dfrac{1}{33}\right)=0\)
=>x-60=0
=>x=60
d: \(\dfrac{x+1}{11}-\dfrac{2x-5}{15}=\dfrac{3x-47}{17}-\dfrac{4x-59}{19}\)
=>\(\dfrac{x+1}{11}+\dfrac{4x-59}{19}=\dfrac{3x-47}{17}+\dfrac{2x-5}{15}\)
=>\(\dfrac{x+1}{11}-1+\dfrac{4x-59}{19}+1=\dfrac{3x-47}{17}+1+\dfrac{2x-5}{15}-1\)
=>\(\dfrac{x-10}{11}+\dfrac{4x-40}{19}=\dfrac{3x-30}{17}+\dfrac{2x-20}{15}\)
=>\(\left(x-10\right)\left(\dfrac{1}{11}+\dfrac{4}{19}-\dfrac{3}{17}-\dfrac{2}{15}\right)=0\)
=>x-10=0
=>x=10
