If $-2 \leq x \leq 5,-3 \leq y \leq 7,4 \leq z \leq 8$, and $w=x y-z$, then the smallest value $w$ may have is
(A) -14
(B) -18
(C) -19
(D) -22
(E) -23
If $-2 \leq x \leq 5,-3 \leq y \leq 7,4 \leq z \leq 8$, and $w=x y-z$, then the smallest value $w$ may have is
(A) -14
(B) -18
(C) -19
(D) -22
(E) -23

## Solution

We obtain the smallest value of $w=x y-z$ by finding the smallest value of the product $x y$ and then subtracting the largest value of $z$.

Since both $x$ and $y$ can take on positive or negative values, the smallest product $x y$ will be negative with one of $x$ and $y$ positive and the other negative. The smallest such product $x y$ is $(5)(-3)=-15$.

Thus, the smallest possible value of $w$ is $-15-8=-23$.

ANSWER: (E)