Question Number 10777 by j.masanja06@gmail.com last updated on 24/Feb/17
![find the range or (ranges) of value x can take for x+6>[2x+3]](Q10777.png)
$$\:\mathrm{find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{or}\:\left(\mathrm{ranges}\right)\:\mathrm{of}\:\mathrm{value}\:\mathrm{x} \\ $$ $$\mathrm{can}\:\mathrm{take}\:\mathrm{for}\:\:\mathrm{x}+\mathrm{6}>\left[\mathrm{2x}+\mathrm{3}\right] \\ $$
Commented bymrW1 last updated on 24/Feb/17
$${do}\:{you}\:{mean}\:{x}+\mathrm{6}>\mid\mathrm{2}{x}+\mathrm{3}\mid\:? \\ $$
Commented byj.masanja06@gmail.com last updated on 25/Feb/17
$$\mathrm{yes}\:! \\ $$
Answered by mrW1 last updated on 24/Feb/17
$${if}\:\mathrm{2}{x}+\mathrm{3}\geqslant\mathrm{0},\:{i}.{e}.\:{x}\geqslant−\frac{\mathrm{3}}{\mathrm{2}} \\ $$ $${x}+\mathrm{6}>\mathrm{2}{x}+\mathrm{3} \\ $$ $${x}<\mathrm{3} \\ $$ $$\Rightarrow−\frac{\mathrm{3}}{\mathrm{2}}\leqslant{x}<\mathrm{3} \\ $$ $$ \\ $$ $${if}\:\mathrm{2}{x}+\mathrm{3}<\mathrm{0},\:{i}.{e}.\:{x}<−\frac{\mathrm{3}}{\mathrm{2}} \\ $$ $${x}+\mathrm{6}>−\mathrm{2}{x}−\mathrm{3} \\ $$ $$\mathrm{3}{x}>−\mathrm{9} \\ $$ $${x}>−\mathrm{3} \\ $$ $$\Rightarrow−\mathrm{3}<{x}<−\frac{\mathrm{3}}{\mathrm{2}} \\ $$ $$ \\ $$ $$\Rightarrow{range}\:{of}\:{x}: \\ $$ $$−\mathrm{3}<{x}<\mathrm{3} \\ $$
Answered by lee last updated on 25/Feb/17
$$\left.{sol}\:\mathrm{1}\right)\left({x}+\mathrm{6}\right)^{\mathrm{2}} >\left(\mathrm{2}{x}+\mathrm{3}\right)^{\mathrm{2}} \\ $$ $$\mathrm{3}{x}^{\mathrm{2}} −\mathrm{27}<\mathrm{0},\:−\mathrm{3}<{x}<\mathrm{3} \\ $$ $$\left.{sol}\:\mathrm{2}\right) \\ $$ $$\left.{i}\right){x}\geqq−\frac{\mathrm{3}}{\mathrm{2}},\:{x}+\mathrm{6}>\mathrm{2}{x}+\mathrm{3} \\ $$ $$−\frac{\mathrm{3}}{\mathrm{2}}\leqq{x}<\mathrm{3} \\ $$ $$\left.{ii}\right){x}<−\frac{\mathrm{3}}{\mathrm{2}},\:{x}+\mathrm{6}>−\mathrm{2}{x}−\mathrm{3} \\ $$ $$−\mathrm{3}<{x}<−\frac{\mathrm{3}}{\mathrm{2}} \\ $$ $$\left.\therefore\left.\:\:{i}\right)\cup{ii}\right)\rightarrow−\mathrm{3}<{x}<\mathrm{3} \\ $$