If $x$ is a number less than -2 , which of the following expressions has the least value?
(A) $x$
(B) $x+2$
(C) $\frac{1}{2} x$
(D) $x-2$
(E) $2 x$
Solution 1

Since the result must be the same for any real number $x$ less than -2 , we substitute $x=-4$ into each of the five expressions:
(A) $x=-4$
(B) $x+2=-2$
(C) $\frac{1}{2} x=-2$
(D) $x-2=-6$
(E) $2 x=-8$

Therefore, $2 x$ is the expression with the least value when $x=-4$ and thus must always be the expression with the least value.

## Solution 2

For any real number $x$, we know that $x-2$ is less than $x$ which is less than $x+2$.

Therefore, neither $x$ nor $x+2$ can be the least of the five values.

For any negative real number $x$, the value of $2 x$ will be less than the value of $\frac{1}{2} x$.

Therefore, $\frac{1}{2} x$ cannot be the least of the five values.

Thus, the least of the five values is either $x-2$ or $2 x$.

When $x<-2$, we know that $2 x-(x-2)=x+2<0$.

Since the difference between $2 x$ and $x-2$ is negative, then $2 x$ has the smaller value and so is the least of all five values.

ANSWER: (E)