Find the values of $x$ which satisfies the inequation $\;$ $-2 \leq \dfrac{1}{2} - \dfrac{2x}{3} \leq 1 \dfrac{5}{6}, \; x \in N$
Graph the solution set on a number line.
Consider $\;$ $-2 \leq \dfrac{1}{2} - \dfrac{2x}{3}$
i.e. $\;$ $-2 \leq \dfrac{3 - 4x}{6}$
i.e. $\;$ $-12 \leq 3 - 4x$
i.e. $\;$ $4x \leq 15$
i.e. $\;$ $x \leq \dfrac{15}{4}$
i.e. $\;$ $x \leq 3.75$ $\;\;\; \cdots \; (1)$
Consider $\;$ $\dfrac{1}{2} - \dfrac{2x}{3} \leq 1 \dfrac{5}{6}$
i.e. $\;$ $\dfrac{1}{2} - \dfrac{2x}{3} \leq \dfrac{11}{6}$
i.e. $\;$ $\dfrac{3 - 4x}{6} \leq \dfrac{11}{6}$
i.e. $\;$ $3 - 4x \leq 11$
i.e. $\;$ $-4x \leq 8$
i.e. $\;$ $-x \leq 2$
i.e. $\;$ $-2 \leq x$ $\;\;\; \cdots \; (2)$
$\therefore \;$ We have from equations $(1)$ and $(2)$, the values of $x$ which satisfy the given inequation as:
$\left\{x \; \big | -2 \leq x \leq 3.75 \right\}$
$\because \;$ $x \in N$, $\;$ the required values of $x$ are: $\;$ $\left\{1, \; 2, \; 3 \right\}$
