1. Jan 2009

2. AC < AB
2x - 4 < x + 6
x - 4 < + 6 (subtract x on both sides)
x < 10 (add 4 to both sides)

3. AC < AE
AC < AD
2x - 4 < 4x - 14
2x - 4 < 3x - 7
-2x - 4 < (subtract 4x on both sides)
-x - 4 < - 7 (subtract 3x on both sides)
-x < -3 (add 4 to both sides)
-2x < -10 (add 4 to both sides)
x > 3 (Inverting the equation by dividing the equation by -1)
x > 5 (Inverting the equation by dividing by - 2)

4. CD < CB and CD < CE
2x – 1 < 3x - 7
2x – 1 < x + 8
-x – 1 < - 7
x – 1 < + 8
-x < - 6
x < 9
x > 6

5. From your previous answers x > 6 but also < 9
Now consider that DE < DB so
2x – 2 < 3x - 9
If x is now > 7, but < 9 the x = 8
-x – 2 < - 9
-x < - 7
x > 7

6. Total = 72 Remember x = 8 AC = 2x – 4 = 12 A → C → D → E → B → A
CD = 2x – 1 = 15 12 15 14 17 14
DE = 2x – 2 = 14
EB = x + 9 = 17
Total = 72
BA = x + 6 = 14

8. 𝐴 →𝐵 →𝐶 →𝐷 →𝐸 →𝐹 →𝐴 𝐹 →𝐷 →𝐶 →𝐴 →𝐵 →𝐸 →𝐹
Note - You can start at any letter and visit every other letter once, in any order, finally returning to your start letter.
𝐹 →𝐷 →𝐶 →𝐴 →𝐵 →𝐸 →𝐹 20 15 5 25 15 15
TOTAL = 95
It is a TOUR that exist, but MAY BE improved upon

9. 𝐹 →𝐸 →𝐶 →𝐴 →𝐵 →𝐷 →𝐹 30 7 5 25 11 10
TOTAL = 88

10. Jan 2010
B → E → C → D → A → B =
12.0 or 12
B → D → A → C → E → B =
13.5

11. The best upper bound is the lowest value, so 12.0
From To
The best upper bound is the lowest value, so 12.0
B → A → D → E → C → B =
12.1

12. June 2010

13. 𝑆 →𝑇 →𝑅 →𝐼 →𝑁 →𝐺 →𝑆 64 70 82 80 82 72
TOTAL = 450
𝑁 →𝐺 →𝑆 →𝑇 →𝑅 →𝐼 →𝑁

14. Using Kruskal or Prim draw a minimum spanning tree
Delete 𝑆
Using Kruskal or Prim draw a minimum spanning tree
Total minimum spanning tree = 𝟐𝟗𝟑 𝑮 𝟕𝟔
Add the two shortest deleted edges 𝑰 𝟕𝟑 𝑵 𝟕𝟒 𝑻 𝑹 𝟕𝟎 𝑰 𝟔𝟖
= 𝟏𝟑𝟐 𝑺 𝑻 𝟔𝟒
Lower Bound = 𝟒𝟐𝟓