Question

Розв’яжіть нерівність ≥ 4х-1.

Answer 1

$(\frac{1}{2} )^{x^{2} +x-2} \geq 4^{x-1}$

$2^{-x^{2} -x+2}\geq 2^{2x-2}$

$-x^{2} -x+2\geq 2x-1$

$-x^{2} +x-2x+2\geq 2(x-1)$

$-x*(x-1)-2*(x-1)\geq 2*(x-1)$

$-(x-1)*(x+2)\geq 2(x -1)$

$-(x-1)*(x+2)-2(x -1)\geq 0$

$-(x-1)*(x+2+2)\geq 0$

$-(x-1)*(x+4)\geq 0$

$\left \{ {{-(x-1)\geq 0} \atop {x+4\geq 0}} \right.$  ⇒$\left \{ {{x\leq 1} \atop {x\geq -4}} \right.$  x ∈ [-4; 1]

$\left \{ {{-(x-1)\leq 0} \atop {x+4\leq 0}} \right.$  ⇒$\left \{ {{x\geq 1} \atop {x\leq -4}} \right.$  x ∈ ∅

ответ: x ∈ [-4; 1]