Question

If $x\in [-3,1),$ then $\frac{1}{x^3}$ belongs to

• A. $(-\infty ,-\frac{1}{9}]\cup (1,\infty )$
• B. $(-\infty ,-\frac{1}{27}]\cup (1,\infty )$
• C. $[-\frac{1}{27},1)$
• D. ​$(-\infty ,-\frac{1}{27}]\cup [1,\infty )$

Answer 1

The correct option is B $(-\infty ,-\frac{1}{27}]\cup (1,\infty )$

$x\in [-3,1)$
Splliting into intervals
$x\in [-3,0)\cup [0,1)$
$\Rightarrow x^3\in [-27,0)\cup [0,1)$
$\Rightarrow x^3\in [-27,1)$
Here $(-27)\left(1\right)<0$
​​​​​​​$\Rightarrow \frac{1}{x^3}\in (-\infty ,-\frac{1}{27}]\cup (1,\infty )$