$$ \begin{aligned} \frac{3}{2}x-\frac{x}{3}+\frac{4}{x^2} &= 0&& \text{multiply ALL terms by } \color{blue}{ 2\cdot3x^2 }. \\[1 em]2\cdot3x^2\frac{3}{2}x-2\cdot3x^2\frac{x}{3}+2\cdot3x^2\cdot\frac{4}{x^2} &= 2\cdot3x^2\cdot0&& \text{cancel out the denominators} \\[1 em]9x-2x+24 &= 0&& \text{simplify left side} \\[1 em]7x+24 &= 0&& \text{ move the constants to the right } \\[1 em]7x &= -24&& \text{ divide both sides by $ 7 $ } \\[1 em]x &= -\frac{24}{7}&& \\[1 em] \end{aligned} $$

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