14. Given the system of inequalities about $x$
$$
\left\{\begin{array}{l}
x < a \\
x > -a
\end{array}\right.
$$

the solution set contains exactly two integers. Find the range of real values for $a$.

14. From the given, we know $\frac{a+2}{2}<x<a+1$.

From the problem, there exists an integer $k$ satisfying
$$
\begin{array}{l}
\left\{\begin{array}{l}
k \leqslant \frac{a+2}{2}<k+1, \\
k+2<a+1 \leqslant k+3
\end{array}\right. \\
\Rightarrow\left\{\begin{array}{l}
2 k-2 \leqslant a<2 k, \\
k+1<a \leqslant k+2 .
\end{array}\right. \\
\text { Then }\left\{\begin{array}{l}
k+1<2 k, \\
2 k-2 \leqslant k+2
\end{array}\right. \\
\Rightarrow 1<k \leqslant 4 \Rightarrow k=2,3,4 .
\end{array}
$$

When $k=2$, we get $3<a<4$;
When $k=3$, we get $4<a \leqslant 5$;
When $k=4$, we get $a=6$.
Therefore, $3<a<4$ or $4<a \leqslant 5$ or $a=6$.