Question

1. Solve each of the following pairs of inequalities​

Answer 1

Answer is given in attachment.

Answer 2

Given Question :-

Solve each of the following pairs of inequalities :

$\sf \: (a) \: \: 5 - a \leqslant a - 6 \leqslant 10 - 3a$

$\sf \: (b) \: \: 4 - b < 2b - 1 < 7 + b$

$\green{\large\underline{\sf{Solution-(a)}}}$

Given inequality is

$\rm :\longmapsto\:\: 5 - a \leqslant a - 6 \leqslant 10 - 3a$

can be rewritten as

$\rm :\longmapsto\:\: 5 - a \leqslant a - 6 \: and \: a - 6\leqslant 10 - 3a$

$\rm :\longmapsto\:\: - a - a \leqslant - 5 - 6 \: and \: a + 3a\leqslant 10 + 6$

$\rm :\longmapsto\:\: - 2a \leqslant - 11 \: and \: 4a\leqslant 16$

$\bf\implies \:a \geqslant \dfrac{11}{2} \: \: and \: \: a \leqslant 4$

$\bf\implies \:x \: \in \: \bigg[\dfrac{11}{2}, \: 4\bigg]$

$\green{\large\underline{\sf{Solution-(b)}}}$

Given inequality is

$\rm :\longmapsto\: 4 - b < 2b - 1 < 7 + b$

can be rewritten as

$\rm :\longmapsto\: 4 - b < 2b - 1 \: \: and \: \: 2b - 1< 7 + b$

$\rm :\longmapsto\: - b - 2b < - 1 - 4 \: \: and \: \: 2b - b < 7 + 1$

$\rm :\longmapsto\: - 3b < - 5 \: \: and \: \: b < 8$

$\rm :\longmapsto\: b > \dfrac{5}{3} \: \: and \: \: b < 8$

$\bf\implies \:\dfrac{5}{3} < b < 8$

$\bf\implies \:b \: \in \: \bigg(\dfrac{5}{3} , \: 8\bigg)$

Additional Information :-

$\boxed{ \bf{ \: x > y \: \: \bf\implies \: - x < - y}}$

$\boxed{ \bf{ \: x < y \: \: \bf\implies \: - x > - y}}$

$\boxed{ \bf{ \: - x < y \: \: \bf\implies \: x > - y}}$

$\boxed{ \bf{ \: - x > y \: \: \bf\implies \: x < - y}}$

$\boxed{ \bf{ \: x \geqslant y \: \: \bf\implies \: - x \leqslant - y}}$

$\boxed{ \bf{ \: - x \geqslant y \: \: \bf\implies \: x \leqslant - y}}$

$\boxed{ \bf{ \: x \leqslant y \: \: \bf\implies \: - x \geqslant - y}}$