Question

-3<-1/2-2x/3<5/6, x€R​

Answer 1

$\large\underline{\sf{Solution-}}$

Given inequality is

$\rm :\longmapsto\: - 3 < - \dfrac{1}{2} - \dfrac{2x}{3} < \dfrac{5}{6}$

can be further rewritten as

$\rm :\longmapsto\: - 3 < - \dfrac{3}{6} - \dfrac{4x}{6} < \dfrac{5}{6}$

On multiply by 6, each term, we get

$\rm :\longmapsto\: - 18 < - 3 - 4x < 5$

On adding 3 in each term, we get

$\rm :\longmapsto\: - 18 + 3 < 3 - 3 - 4x < 5 + 3$

$\rm :\longmapsto\: - 15 < - 4x < 8$

On dividing each term by - 4, we get

$\rm :\longmapsto\:\dfrac{15}{4} > x > - 2$

can be further rewritten as

$\rm :\longmapsto\: - 2 < x < \dfrac{15}{4}$

$\bf\implies \:x \: \in \: \bigg( - 2, \: \dfrac{15}{4} \bigg)$

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

More to know

$\green{\boxed{\tt{ x > y \: \rm\implies \: - x < - y}}}$

$\green{\boxed{\tt{ - x > y \: \rm\implies \: x < - y}}}$

$\green{\boxed{\tt{ x \geqslant y \: \rm\implies \: - x \leqslant - y}}}$

$\green{\boxed{\tt{ x \geqslant - y \: \rm\implies \: - x \leqslant y}}}$

$\red{\boxed{\tt{ |x| < y \: \: \rm\implies \: - y < x < y \: }}}$

$\red{\boxed{\tt{ |x| \leqslant y \: \: \rm\implies \: - y \leqslant x \leqslant y \: }}}$

$\red{\boxed{\tt{ |x - z| \leqslant y \: \: \rm\implies \: z- y \leqslant x \leqslant y + z \: }}}$

$\red{\boxed{\tt{ |x - z| < y \: \: \rm\implies \: z- y < x < y + z \: }}}$

Answer 2

$\large\underline{\sf{Solution-}}$

Given inequality is

$\rm :\longmapsto\: - 3 < - \dfrac{1}{2} - \dfrac{2x}{3} < \dfrac{5}{6}$

can be further rewritten as

$\rm :\longmapsto\: - 3 < - \dfrac{3}{6} - \dfrac{4x}{6} < \dfrac{5}{6}$

On multiply by 6, each term, we get

$\rm :\longmapsto\: - 18 < - 3 - 4x < 5$

On adding 3 in each term, we get

$\rm :\longmapsto\: - 18 + 3 < 3 - 3 - 4x < 5 + 3$

$\rm :\longmapsto\: - 15 < - 4x < 8$

On dividing each term by - 4, we get

$\rm :\longmapsto\:\dfrac{15}{4} > x > - 2$

can be further rewritten as

$\rm :\longmapsto\: - 2 < x < \dfrac{15}{4}$

$\bf\implies \:x \: \in \: \bigg( - 2, \: \dfrac{15}{4} \bigg)$

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

More to know

$\green{\boxed{\tt{ x > y \: \rm\implies \: - x < - y}}}$

$\green{\boxed{\tt{ - x > y \: \rm\implies \: x < - y}}}$

$\green{\boxed{\tt{ x \geqslant y \: \rm\implies \: - x \leqslant - y}}}$

$\green{\boxed{\tt{ x \geqslant - y \: \rm\implies \: - x \leqslant y}}}$

$\red{\boxed{\tt{ |x| < y \: \: \rm\implies \: - y < x < y \: }}}$

$\red{\boxed{\tt{ |x| \leqslant y \: \: \rm\implies \: - y \leqslant x \leqslant y \: }}}$

$\red{\boxed{\tt{ |x - z| \leqslant y \: \: \rm\implies \: z- y \leqslant x \leqslant y + z \: }}}$

$\red{\boxed{\tt{ |x - z| < y \: \: \rm\implies \: z- y < x < y + z \: }}}$