7 Liam is taking part in a treasure hunt. There are five clues to be solved and they are at the points $A , B , C , D$ and $E$. The table shows the distances between pairs of points. All of the distances are functions of $x$, where $\boldsymbol { x }$ is an integer.

Liam must travel to all five points, starting and finishing at $A$.

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
 & $\boldsymbol { A }$ & $\boldsymbol { B }$ & $\boldsymbol { C }$ & $\boldsymbol { D }$ & $\boldsymbol { E }$ \\
\hline
A & - & $x + 6$ & $2 x - 4$ & $3 x - 7$ & $4 x - 14$ \\
\hline
$\boldsymbol { B }$ & $x + 6$ & - & $3 x - 7$ & $3 x - 9$ & $x + 9$ \\
\hline
$\boldsymbol { C }$ & $2 x - 4$ & $3 x - 7$ & - & $2 x - 1$ & $x + 8$ \\
\hline
$\boldsymbol { D }$ & $3 x - 7$ & $3 x - 9$ & $2 x - 1$ & - & $2 x - 2$ \\
\hline
E & $4 x - 14$ & $x + 9$ & $x + 8$ & $2 x - 2$ & - \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item The nearest point to $A$ is $C$.
\begin{enumerate}[label=(\roman*)]
\item By considering $A C$ and $A B$, show that $x < 10$.
\item Find two other inequalities in $x$.
\end{enumerate}\item The nearest neighbour algorithm, starting from $A$, gives a unique minimum tour $A C D E B A$.
\begin{enumerate}[label=(\roman*)]
\item By considering the fact that Liam's tour visits $D$ immediately after $C$, find two further inequalities in $x$.
\item Find the value of the integer $x$.
\item Hence find the total distance travelled by Liam if he uses this tour.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{AQA D1 2009 Q7 [13]}}