Question

The value of $x$ which satisfies the inequalities $\frac{x-3}{4}-x<\frac{x-1}{2}-\frac{x-2}{3}$ and $2-x>2x-8.$

• A. $x<-1$ and $x<\frac{10}{3}$
• B. $x<11$ and $x<\frac{10}{3}$
  
• C. $x<1$ and $x<-\frac{10}{3}$
  
• D. none of these

Answer 1

Answer: (A)

$x<-1$ and $x<\frac{10}{3}$

$\frac{x-3}{4}-x<\frac{x-1}{2}-\frac{x-2}{3}$ and $2-x>2x-8$

$\Rightarrow \frac{x-3-4x}{4}<\frac{3x-3-2x+4}{6}$ and $\Rightarrow -x-2x>-8-2$

$\Rightarrow \frac{-3-3x}{4}<\frac{x+1}{6}$ and $\Rightarrow -3x>-10$

$\Rightarrow \frac{-3-3x}{2}<\frac{x+1}{3}$ and $\Rightarrow x<\frac{10}{3}$

$\Rightarrow -9-9x<2x+2$ and $\Rightarrow x<\frac{10}{3}$

$\Rightarrow -9x-2x>2+9$ and $\Rightarrow x<\frac{10}{3}$

$\Rightarrow -11x>11$ and $\Rightarrow x<\frac{10}{3}$

$\Rightarrow x<-1$ and $\Rightarrow x<\frac{10}{3}$