Question
Solve the following inequation, write the solution set and represent it on the number line:
$-\frac{x}{3} \leq \frac{x}{2}-1 \frac{1}{3}<\frac{1}{6}, x \in R$
$-\frac{x}{3} \leq \frac{x}{2}-1 \frac{1}{3}<\frac{1}{6}, x \in R$
✓
Answer
$-\frac{x}{3} \leq \frac{x}{2}-1 \frac{1}{3}<\frac{1}{6}, x \in R $
$ -\frac{x}{3} \leq \frac{x}{2}-1 \frac{1}{3}$
$ -\frac{x}{3} \leq \frac{x}{2}-\frac{4}{3}$
$\frac{4}{3} \leq \frac{x}{2}+\frac{x}{3}$
$ \frac{4}{3} \leq \frac{5 x}{6} $
$ \frac{6}{5} \times \frac{4}{3} \leq x$
$\frac{8}{5} \leq x $
$ \frac{x}{2}-1 \frac{1}{3}<\frac{1}{6} $
$\frac{x}{2}<\frac{1}{6}+\frac{4}{3} $
$ \frac{x}{2}<\frac{1+8}{6}$
$x<\frac{9 \times 2}{6}$
$x < 3$
From $(1)$ and $(2)$
$\frac{8}{5} \leq x<3$
or $1·6 \leq x < 3$
$\therefore$ Solution set $\{x : 1·6 \leq x < 3, x \in R\}$
$ -\frac{x}{3} \leq \frac{x}{2}-1 \frac{1}{3}$
$ -\frac{x}{3} \leq \frac{x}{2}-\frac{4}{3}$
$\frac{4}{3} \leq \frac{x}{2}+\frac{x}{3}$
$ \frac{4}{3} \leq \frac{5 x}{6} $
$ \frac{6}{5} \times \frac{4}{3} \leq x$
$\frac{8}{5} \leq x $
$ \frac{x}{2}-1 \frac{1}{3}<\frac{1}{6} $
$\frac{x}{2}<\frac{1}{6}+\frac{4}{3} $
$ \frac{x}{2}<\frac{1+8}{6}$
$x<\frac{9 \times 2}{6}$
$x < 3$
From $(1)$ and $(2)$
$\frac{8}{5} \leq x<3$
or $1·6 \leq x < 3$
$\therefore$ Solution set $\{x : 1·6 \leq x < 3, x \in R\}$
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