$$-\frac{7}{9}(x+3) = 14$$

$$ \begin{aligned} -\frac{7}{9}(x+3) &= 14&& \text{multiply ALL terms by } \color{blue}{ 9 }. \\[1 em]-9 \cdot \frac{7}{9}(x+3) &= 9\cdot14&& \text{cancel out the denominators} \\[1 em]-(7x+21) &= 126&& \text{simplify left side} \\[1 em]-7x-21 &= 126&& \text{ move -21 to the right } \\[1 em]-7x &= 126+21&& \\[1 em]-7x &= 147&& \text{ divide both sides by $ -7 $ } \\[1 em]x &= -\frac{147}{7}&& \\[1 em]x &= -21&& \\[1 em] \end{aligned} $$

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