Answer
$\text{False}$
Work Step by Step
The multiplication property of inequalities states that if we multiply an inequality by a negative number, then we reverse the direction of the inequality.
Since $c\lt 0$, then $\frac{1}{c}\lt 0$.
Thus, if $a\lt b$ and $x\lt0$, then
$$\frac{1}{c} \cdot a \gt \frac{1}{c} \cdot b \implies\frac{a}{c}\gt\frac{b}{c}$$
Therefore, the given statement is false.
