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Question Number 170053 by mathlove last updated on 15/May/22

Answered by greougoury555 last updated on 15/May/22

((6x)/(x−2)) −(√((12x)/(x−2)))−2 (((12x)/(x−2)))^(1/4) ≥ 0 let (((12x)/(x−2)))^(1/4) = u ; u≥0 ⇒ (1/2)u^4 −u^2 −2u ≥ 0 ⇒u^4 −2u^2 −4u ≥ 0 ⇒u(u^3 −2u−4) ≥ 0 ⇒u(u−2)(u^2 +2u+2)_(>0) ≥0 ⇒u≥2 or u≤0 (rejected) ⇒(((12x)/(x−2)))^(1/4) ≥ 2 ; ((12x)/(x−2)) ≥ 16 ⇒((3x)/(x−2)) −((4(x−2))/(x−2)) ≥ 0 ⇒((8−x)/(x−2)) ≥0 ; ((x−8)/(x−2)) ≤ 0 ; 2<x≤8

6xx−2−12xx−2−212xx−24⩾0let12xx−24=u;u⩾0⇒12u4−u2−2u⩾0⇒u4−2u2−4u⩾0⇒u(u3−2u−4)⩾0⇒u(u−2)(u2+2u+2)⏟>0⩾0⇒u⩾2oru⩽0(rejected)⇒12xx−24⩾2;12xx−2⩾16⇒3xx−2−4(x−2)x−2⩾0⇒8−xx−2⩾0;x−8x−2⩽0;2<x⩽8

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