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Question Number 984 by 112358 last updated on 13/May/15
Show that ((x(x+1))/(3x−1))>1 given that x>(1/3) .
Showthatx(x+1)3x−1>1giventhatx>13.
Commented by prakash jain last updated on 13/May/15
For x=1 LHS=((1×2)/2)=1≯1 So the inequality should be ((x(x+1))/(3x−1))≥1
Forx=1LHS=1×22=1≯1Sotheinequalityshouldbex(x+1)3x−1⩾1
Answered by prakash jain last updated on 13/May/15
(x−1)^2 ≥0⇒x^2 −2x+1≥0 ⇒x^2 +x−3x+1≥0 ⇒x(x+1)≥3x−1 x>(1/3)⇒3x−1>0, divide by 3x−1 hencd ((x(x+1))/(3x−1))≥1
(x−1)2⩾0⇒x2−2x+1⩾0⇒x2+x−3x+1⩾0⇒x(x+1)⩾3x−1x>13⇒3x−1>0,divideby3x−1hencdx(x+1)3x−1⩾1

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