Question Number 67687 by Rio Michael last updated on 30/Aug/19
$${find}\:{the}\:{range}\:{of}\:{values}\:{of}\: \\ $$$$\:\:\mid\frac{{x}^{\mathrm{2}} −\mathrm{9}}{\mathrm{3}_{\:} }\mid=\:\frac{\mathrm{9}−{x}^{\mathrm{2}} }{\mathrm{3}} \\ $$$$ \\ $$
Commented by mr W last updated on 30/Aug/19
$${LHS}\geqslant\mathrm{0} \\ $$$$\Rightarrow\mathrm{9}−{x}^{\mathrm{2}} \geqslant\mathrm{0} \\ $$$$\Rightarrow{x}^{\mathrm{2}} \leqslant\mathrm{9} \\ $$$$\Rightarrow−\mathrm{3}\leqslant{x}\leqslant\mathrm{3} \\ $$$$\mathrm{9}−{x}^{\mathrm{2}} \leqslant\mathrm{9} \\ $$$$\frac{\mathrm{9}−{x}^{\mathrm{2}} }{\mathrm{3}}\leqslant\mathrm{3} \\ $$$$\Rightarrow\mathrm{0}\leqslant\frac{\mathrm{9}−{x}^{\mathrm{2}} }{\mathrm{3}}\leqslant\mathrm{3} \\ $$
Commented by Rio Michael last updated on 30/Aug/19
$${sir}\:{is}\:{it}\:\:{LHS}\:\:<\mathrm{0}\:{or}\:{LHS}\:\geqslant\mathrm{0}? \\ $$
Commented by mr W last updated on 30/Aug/19
$${LSH}=\mid...\mid\geqslant\mathrm{0} \\ $$