Question

Solve for $x:|x-|3-x||-3x=8$

• A. $\{-1\}$
• B. $\left\{1\right\}$
• C. $\{-\frac{3}{2}\}$
• D. $\left\{\frac{3}{2}\right\}$

Answer 1

The correct option is A $\{-1\}$

$|x-|3-x||-3x=8$

Case$-1:$
$x>3$
$\Rightarrow |x-(-(3-x))|-3x=8$
$\Rightarrow |x+3-x|-3x=8$
$\Rightarrow x=-\frac{5}{3}$
But $x$ shall be $>3$
So, $x\in \varphi$

Case$-2:$
$x\le 3$
$\Rightarrow |x-(3-x)|-3x=8$
$\Rightarrow |2x-3|-3x=6$
Case$-2a:$
$x<\frac{3}{2}$
$\Rightarrow x\in (-\infty ,\frac{3}{2})$
$\Rightarrow -(2x-3)-3x=8$
$\Rightarrow -5x=5$
$\Rightarrow x=-1\in (-\infty ,\frac{3}{2})$

Case$-2b:$
$x\ge \frac{3}{2}$
$x\in [\frac{3}{2},3]$
$\Rightarrow (2x-3)-3x=8$
$\Rightarrow -x=11$
$\Rightarrow x=-11\notin [\frac{3}{2},3]$

Final solution $=$ case $1\cup$ case $2$
$=\varphi \cup \{-1\}=\{-1\}$