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Given-p-lt-x-lt-q-is-solution-set-inequality-1-2-x-2-2x-2-3x-gt-2-for-x-1-find-the-value-of-5p-3q-




Question Number 142968 by bramlexs22 last updated on 08/Jun/21
Given p<x<q is solution set  inequality 1+2^x +2^(2x) +2^(3x) +...>2  for x≠1. find the value of   5p−3q .
$${Given}\:{p}<{x}<{q}\:{is}\:{solution}\:{set} \\ $$$${inequality}\:\mathrm{1}+\mathrm{2}^{{x}} +\mathrm{2}^{\mathrm{2}{x}} +\mathrm{2}^{\mathrm{3}{x}} +…>\mathrm{2} \\ $$$${for}\:{x}\neq\mathrm{1}.\:{find}\:{the}\:{value}\:{of}\: \\ $$$$\mathrm{5}{p}−\mathrm{3}{q}\:. \\ $$
Answered by mr W last updated on 08/Jun/21
−1<2^x <1 ⇒x<0  (1/(1−2^x ))>2 ⇒2^x >2^(−1)  ⇒x>−1  ⇒−1<x<0  i.e. p=−1, q=0  5p−3q=−5
$$−\mathrm{1}<\mathrm{2}^{{x}} <\mathrm{1}\:\Rightarrow{x}<\mathrm{0} \\ $$$$\frac{\mathrm{1}}{\mathrm{1}−\mathrm{2}^{{x}} }>\mathrm{2}\:\Rightarrow\mathrm{2}^{{x}} >\mathrm{2}^{−\mathrm{1}} \:\Rightarrow{x}>−\mathrm{1} \\ $$$$\Rightarrow−\mathrm{1}<{x}<\mathrm{0} \\ $$$${i}.{e}.\:{p}=−\mathrm{1},\:{q}=\mathrm{0} \\ $$$$\mathrm{5}{p}−\mathrm{3}{q}=−\mathrm{5} \\ $$

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