Question

Solve the following inequation, write the solution set and represent it on the number line: $-\frac{x}{3}\le \frac{x}{2}-1\frac{1}{3}<\frac{1}{6},x\in R$.

Answer 1

Given: $-\frac{x}{3}\le \frac{x}{2}-1\frac{1}{3}<\frac{1}{6}$
Taking L.C.M. of $3,2$ and $6$ is $6$.
$-\frac{x}{3}\times 6\le \frac{x}{2}\times 6-\frac{4}{3}\times 6<\frac{1}{6}\times 6$
$-2x\le 3x-8<1$
$\Rightarrow -2x\le 3x-8$ and $3x-8<1$
$\Rightarrow 8\le 3x+2x$ $\Rightarrow 3x<1+8$
$\Rightarrow 8\le 5x$ $\Rightarrow 3x<9$
$\Rightarrow \frac{8}{5}\le x$ $\Rightarrow x<3$
$\therefore$ The solution set is $\{x:1.6\le x\le 3,xϵR\}$