\section*{Exercise 3 - 171023}

Determine the set of all real numbers \(x\) for which the term \(\lg \left(x^{2}+7 x-30\right)\) is defined!
}

The term \(\lg \left(x^{2}+7 x-30\right)\) is defined if and only if \(x^{2}+7 x-30>0\). To this end, the following are equivalent in sequence

\[
\begin{gathered}
x^{2}+7 x+\left(\frac{7}{2}\right)^{2}>30+\frac{49}{4} \\
\left(x+\frac{7}{2}\right)^{2}>\frac{169}{4} \\
\left|x+\frac{7}{2}\right|>\frac{13}{2} \\
x+\frac{7}{2}>\frac{13}{2} \quad \text { or } \quad x+\frac{7}{2}<-\frac{13}{2} \\
x>3 \quad \text { or } \quad x<-10
\end{gathered}
\]

Thus, the term \(\lg \left(x^{2}+7 x-30\right)\) is defined if \(x>3\) or \(x<-10\).