$$1+2 \cdot \frac{x}{3}+4-\frac{1}{7} = \frac{1}{9}$$

$$ \begin{aligned} 1+2 \cdot \frac{x}{3}+4-\frac{1}{7} &= \frac{1}{9}&& \text{multiply ALL terms by } \color{blue}{ 63 }. \\[1 em]63\cdot1+63\cdot2 \cdot \frac{x}{3}+63\cdot4-63\cdot\frac{1}{7} &= 63\cdot\frac{1}{9}&& \text{cancel out the denominators} \\[1 em]63+42x+252-9 &= 7&& \text{simplify left side} \\[1 em]42x+306 &= 7&& \text{ move 306 to the right } \\[1 em]42x &= 7-306&& \\[1 em]42x &= -299&& \text{ divide both sides by $ 42 $ } \\[1 em]x &= -\frac{299}{42}&& \\[1 em] \end{aligned} $$

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