Question

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Answer 1

$Domain:\ x\neq0\\\\\dfrac{1}{x}+\dfrac{1}{3x^2}=\dfrac{1}{3x}\\\\\dfrac{1\cdot3x}{x\cdot3x}+\dfrac{1}{3x^2}=\dfrac{1}{3x}\\\\\dfrac{3x}{3x^2}+\dfrac{1}{3x^2}=\dfrac{1}{3x}\\\\\dfrac{3x+1}{3x^2}=\dfrac{1}{3x}\qquad\text{cross multiply}\\\\(3x)(3x+1)=(3x^2)(1)\qquad\text{use distributive property}\\\\(3x)(3x)+(3x)(1)=3x^2\\\\9x^2+3x=3x^2\qquad\text{subtract}\ 3x^2\ \text{from both sides}\\\\6x^2+3x=0\\\\3x(2x+1)=0\iff3x=0\ \vee\ 2x+1=0\\\\3x=0\qquad\text{divide both sides by 3}\\x=0\notin Domain$

$2x+1=0\qquad\text{subtract 1 from both sides}\\\\2x=-1\qquad\text{divide both sides by 2}\\\\\boxed{x=-\dfrac{1}{2}}\to\boxed{A)}$