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3\left(x+1\right)-5\left(x-3\right)=0
Multiply both sides of the equation by 30, the least common multiple of 10,6.
3x+3-5\left(x-3\right)=0
Use the distributive property to multiply 3 by x+1.
3x+3-5x+15=0
Use the distributive property to multiply -5 by x-3.
-2x+3+15=0
Combine 3x and -5x to get -2x.
-2x+18=0
Add 3 and 15 to get 18.
-2x=-18
Subtract 18 from both sides. Anything subtracted from zero gives its negation.
x=\frac{-18}{-2}
Divide both sides by -2.
x=9
Divide -18 by -2 to get 9.