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