The median of the set of numbers $\{$12, 38, 45, $x$, 14$\}$ is five less than the mean. If $x$ is a negative integer, what is the value of $x$?
To find the median of a set of numbers, I need to arrange them in order from least to greatest and find the middle value. If $x$ is a negative integer, then it must be the least value in the set, so the order is $x$, 12, 14, 38, 45. The median is the third value, which is 14. The mean is the sum of all the values divided by the number of values, which is 5 in this case. So the mean is $(x + 12 + 14 + 38 + 45) \div 5$. The problem says that the median is five less than the mean, so I can write an equation: 14 = mean - 5. Substituting the expression for the mean, I get 14 = $(x + 12 + 14 + 38 + 45) \div 5 - 5$. Multiplying both sides by 5, I get 70 = $x + 12 + 14 + 38 + 45 - 25$. Simplifying, I get 70 = $x + 84$. Subtracting 84 from both sides, I get $x = -14$. This is a negative integer, so it is a valid value for $x$. 