Question

Is $\frac{4}{x}<-\frac{1}{3}$, what is the possible range of values for $x$?

• A. $-10<x<0$.
• B. $-12<x<0$.
• C. $2<x<0$.
• D. $0<x<12$.

Answer 1

Answer: (B)

$-12<x<0$.

For this type problem, you have to consider two possibilities: $x$ could be positive or negative. When you multiply the inequality by $x$, you will need to flip the sign when $x$ is negative, but not flip the sign when $x$ is positive. However, notice that $x$ cannot be positive the left hand side of the inequality is less than $-\frac{1}{3}$, which means $\frac{4}{3}$ must be negative. There, $x$ must be negative.

Case 1: $x>0$

Not possible

Case 2: $x<0$

$\frac{4}{3}<-\frac{1}{3}$

$12>-x$ (flip the sign because $x$ is negative)

$-12<x$ (divide by $-1$, flip it again)

Case 1 is not possible

For Case 2, $x$ must be negative AND greater than $-12$. Thus, $-12<x<0$.