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Question Number 77183 by jagoll last updated on 04/Jan/20

what is x   satisfy inequality   3^x^2  × 5^(x−1)  ≥ 3

$$\mathrm{what}\:\mathrm{is}\:\mathrm{x}\: \\ $$$$\mathrm{satisfy}\:\mathrm{inequality}\: \\ $$$$\mathrm{3}^{\mathrm{x}^{\mathrm{2}} } ×\:\mathrm{5}^{\mathrm{x}−\mathrm{1}} \:\geqslant\:\mathrm{3} \\ $$

Answered by john santu last updated on 04/Jan/20

⇒ 3^x^2   × 3^(log_3 (5^(x−1) ))  ≥ 3  3^(x^2 +log_3 (5^(x−1) ))  ≥ 3  x^2 +log_3 (5^(x−1) )−1≥0  (x−1)(x+1)+(x−1)log_3 (5) ≥ 0  (x−1){x+log_3 (15)}≥0  ⇒ x≥ 1 ∨ x ≤ −log_3 (15)

$$\Rightarrow\:\mathrm{3}^{{x}^{\mathrm{2}} } \:×\:\mathrm{3}^{\mathrm{log}_{\mathrm{3}} \left(\mathrm{5}^{{x}−\mathrm{1}} \right)} \:\geqslant\:\mathrm{3} \\ $$$$\mathrm{3}^{{x}^{\mathrm{2}} +\mathrm{log}_{\mathrm{3}} \left(\mathrm{5}^{{x}−\mathrm{1}} \right)} \:\geqslant\:\mathrm{3} \\ $$$${x}^{\mathrm{2}} +\mathrm{log}_{\mathrm{3}} \left(\mathrm{5}^{{x}−\mathrm{1}} \right)−\mathrm{1}\geqslant\mathrm{0} \\ $$$$\left({x}−\mathrm{1}\right)\left({x}+\mathrm{1}\right)+\left({x}−\mathrm{1}\right)\mathrm{log}_{\mathrm{3}} \left(\mathrm{5}\right)\:\geqslant\:\mathrm{0} \\ $$$$\left({x}−\mathrm{1}\right)\left\{{x}+\mathrm{log}_{\mathrm{3}} \left(\mathrm{15}\right)\right\}\geqslant\mathrm{0} \\ $$$$\Rightarrow\:{x}\geqslant\:\mathrm{1}\:\vee\:{x}\:\leqslant\:−\mathrm{log}_{\mathrm{3}} \left(\mathrm{15}\right) \\ $$

Commented by jagoll last updated on 04/Jan/20

thanks sir

$$\mathrm{thanks}\:\mathrm{sir} \\ $$

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