\(\Rightarrow\left(\dfrac{x+5}{100}+1\right)+\left(\dfrac{x+4}{101}+1\right)=\left(\dfrac{x+3}{102}+1\right)+\left(\dfrac{x+2}{103}+1\right)\\ \Rightarrow\dfrac{x+105}{100}+\dfrac{x+105}{101}=\dfrac{x+105}{102}+\dfrac{x+105}{103}\\ \Rightarrow\left(x+105\right)\left(\dfrac{1}{100}+\dfrac{1}{101}-\dfrac{1}{102}-\dfrac{1}{103}\right)=0\\ \Rightarrow x=-105\left(\dfrac{1}{100}+\dfrac{1}{101}-\dfrac{1}{102}-\dfrac{1}{103}\ne0\right)\)
\(\dfrac{x+5}{100}+\dfrac{x+4}{101}=\dfrac{x+3}{102}+\dfrac{x+2}{103}\)
<=> \(\left(\dfrac{x+5}{100}+1\right)+\left(\dfrac{x+4}{101}+1\right)=\left(\dfrac{x+3}{102}+1\right)+\left(\dfrac{x+2}{103}+1\right)\)
<=> \(\dfrac{x+105}{100}+\dfrac{x+105}{101}-\dfrac{x+105}{102}-\dfrac{x+105}{103}=0\)
<=> \(\left(x+105\right)\left(\dfrac{1}{100}+\dfrac{1}{101}-\dfrac{1}{102}-\dfrac{1}{103}\right)=0\)
<=> x + 105 = 0
<=> x = -105
\(\dfrac{x+5}{100}+\dfrac{x+4}{101}=\dfrac{x+3}{102}+\dfrac{x+2}{103}\)
\(\left(\dfrac{x+5}{100}+1\right)+\left(\dfrac{x+4}{101}+1\right)=\left(\dfrac{x+3}{102}+1\right)+\left(\dfrac{x+2}{103}+1\right)\)
\(\dfrac{x+105}{100}+\dfrac{x+105}{101}=\dfrac{x+105}{102}+\dfrac{x+105}{103}\)
\(\Rightarrow\dfrac{x+105}{100}+\dfrac{x+105}{101}-\dfrac{x+105}{102}-\dfrac{x+105}{103}=0\)
\(\Rightarrow\left(x+105\right)\left(\dfrac{1}{100}+\dfrac{1}{101}-\dfrac{1}{102}-\dfrac{1}{103}\right)=0\)
\(\Rightarrow x+105=0\Rightarrow x=-105\)
