Determine if a relation is transitive. Decide if the following relation on X is transitive: X = Z, with relation a R b if −7 ≤ a − b ≤ 7
By adding
−7 ≤ a − b ≤ 7 and −7 ≤ b − c ≤ 7
by parts, I found 
−14 ≤ a − c ≤ 14
, which gives a clue but is not enough since I need to prove (or disprove) that 
−7 ≤ a − c ≤ 7... any ideas? Thanks in advance!

In order to show a relation $R$ is transitive, we must show $aRb$ and $bRc$ implies $aRc$. 
Your relation $R$ is equivalent to $aRb$ if $|a - b| \leq 7$, which is not transitive.
Suppose $x = 10, y = 3, z = 0$. Then, we have $xRy$ because $|x - y| = 7 \leq 7$. Also, we have $yRz$ because $|y - z| = 3 \leq 7$. However, we do not have $xRz$ because $|x - z| = 10 > 7.$
Thus, we have a contadiction.
