The first inequality given is:
\[ 7x + 3 < 4 \]
Subtract 3 from both sides:
\[ 7x < 1 \]
Now, divide both sides by 7:
\[ x < \frac{1}{7} \]
The second inequality given is:
\[ x + 9 > 13 \]
Subtract 9 from both sides:
\[ x > 4 \]
The solution to the compound inequality is:
\[ x < \frac{1}{7} \text{ or } x > 4 \]
\[ \boxed{x < \frac{1}{7} \text{ or } x > 4} \]
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