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\left(b+1\right)\left(-3+1\right)+\left(b+9\right)\left(-\frac{1}{3}+1\right)=0
Variable b cannot be equal to any of the values -9,-1 since division by zero is not defined. Multiply both sides of the equation by \left(b+1\right)\left(b+9\right), the least common multiple of 9+b,1+b.
\left(b+1\right)\left(-2\right)+\left(b+9\right)\left(-\frac{1}{3}+1\right)=0
Add -3 and 1 to get -2.
-2b-2+\left(b+9\right)\left(-\frac{1}{3}+1\right)=0
Use the distributive property to multiply b+1 by -2.
-2b-2+\left(b+9\right)\times \frac{2}{3}=0
Add -\frac{1}{3} and 1 to get \frac{2}{3}.
-2b-2+\frac{2}{3}b+6=0
Use the distributive property to multiply b+9 by \frac{2}{3}.
-\frac{4}{3}b-2+6=0
Combine -2b and \frac{2}{3}b to get -\frac{4}{3}b.
-\frac{4}{3}b+4=0
Add -2 and 6 to get 4.
-\frac{4}{3}b=-4
Subtract 4 from both sides. Anything subtracted from zero gives its negation.
b=-4\left(-\frac{3}{4}\right)
Multiply both sides by -\frac{3}{4}, the reciprocal of -\frac{4}{3}.
b=3
Multiply -4 and -\frac{3}{4} to get 3.