# Problem of the Day

Let and represent positive integers. Is ?

(1)

(2) |e|>|cd|

A. Statement 1 alone is sufficient, but statement 2 alone is not sufficient.

See the Solution

### Solution

[latexpage]

Remember that absolute values can never be negative, so if we can show that $cd-e$ is negative, that will be sufficient to answer the question negatively.

Does $|ab+c|=cd-e$?

Statement 1: SUFFICIENT. This tells us that $cd-e$ is definitely negative. $\sqrt{x^2}=x$ whenever $x\geq0$ but never when $x<0$. Thus, $cd-e$ is negative and cannot equal $|ab+c|$.

Statement 2: INSUFFICENT. $|e|>|cd|$ doesn’t tell us anything important. This would be true if $e=-8$ and $cd=2$ in which case $cd-e$ would equal $10$ or if $cd=-2$ and $e=8$ in which case $cd-e$ would equal $-10$. With this information, $cd-e$ could be positive or negative, plus we still don’t know anything about $ab+c$, so we can’t determine if the expressions are equal.