Question

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

• A. $x<3$ OR $x>4$
• B. $x<0$ OR $x>12$
• C. $x<3$ OR $x>0$
• D. $x<0$ OR $x>3$

Answer 1

The correct option is B $x<0$ OR $x>12$

For this type of problem you have t 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:

Case 1: $x>0$

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

$12<x$

Case 2: $x<0$

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

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

For Case 1, $x$ must be positive AND greater than $12$. Thus $x>12$

For Case 2: $x$ must be negative AND less than $12$. Thus $x<0$

Combined, $x<0$ OR $x>12$