Question

All real numbers x which satisfy the inequality $|1+4i-2^{-x}|\le 5$ where $i=\sqrt{-1},xϵR$ are.

• A. $[-2,\infty )$
• B. $(-\infty ,2]$
• C. $[0,\infty )$
• D. $[-2,0]$

Answer 1

The correct option is A $[-2,\infty )$

Solving the modulus of complex, we get
$=\sqrt{(1-2^{-x}{)}^2+4^2}⩽5$

$=(1-2^{-x}{)}^2+16⩽25$

$=(1-2^{-x}{)}^2⩽9$

$=-3⩽1-2^{-x}⩽3$

$=-4⩽-2^{-x}$

$=-2^2⩽-2^{-x}$

$=x⩾-2$

$=xϵ[-2,\infty )$