1. For some real number $c$, the graphs of the equation $y=|x-20|+|x+18|$ and the line $y=x+c$ intersect at exactly one point. What is $c$ ?
Answer:
18

We want to know the value of $c$ so that the graph $|x-20|+|x+18|-x=c$ has one solution. The graph of the function $|x-20|+|x+18|-x$ consists of an infinite section of slope -3 for $x \in(-\infty,-18]$, then a finite section of slope -1 for $x \in[-18,20]$, then an infinite section of slope 1 for $x \in[20, \infty)$. Notice that this graph is strictly decreasing on $(-\infty, 20]$ and strictly increasing on $[20, \infty)$. Therefore any horizontal line will intersect this graph 0 or 2 times, except the one that passes through the "vertex" $(20,|20-20|+|20+18|-20)=(20,18)$, giving a value of $c=18$.