Question

Решить неравенство log_2(x^2-6x+24)<4

Answer 1

{x²-6x+24>0⇒(x-4)(x-6)>0⇒x<4 U x>6
{x²-6x+24<16⇒x²-6x+8<0⇒(x-4)(x-2)<0⇒2<x<4
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\                       /////////////  
--------(2)--------------(4)--------------(6)----------
         ////////////////////////
x∈(2;4)

Answer 2

$log _{2}( x^{2} -6x+24)\ \textless \ 4\\\\ x^{2} -6x+24\ \textless \ 16\\\\ x^{2} -6x+8\ \textless \ 0\\\\(x-4)(x-2)\ \textless \ 0$
      +                  -                      +
________₀__________₀_________
               2                    4
x ∈ (2 ; 4)
ОДЗ: x² - 6x + 24 > 0
D = 36 - 4 * 1 * 24 = 36 - 96 = - 60
D < 0 , старший коэффициент равен 1 > 0 , значит
x² - 6x + 24 > 0 при любых значениях x.
Ответ: x ∈ (2 ; 4)