Question

Решите неравенство
log_x^2(x−4)2 ≤ 0

Answer 1

Метод рационализации.

$log_{x^2}(x-4)^2leq 0\\log_{x^2}(x-4)^2-log_{x^2}1leq 0 ; ;; Leftrightarrow ; ; left{begin{array}{l}x^2>0; ,; x^2ne 1\(x-4)^2>0\((x-4)^2-1)(x^2-1)leq 0end{array}right\\\(x^2-8x+15)(x-1)(x+1)leq 0\\(x-3)(x-5)(x-1)(x+1)leq 0\\znaki:; ;+++[-1, ]---[, 1, ]+++[, 3, ]---[, 5, ]+++\\xin [-1,1, ]cup [, 3,5, ]\\left{begin{array}{ccc}xne 0; ,; xne pm 1\xne 4\xin [-1,1, ]cup [, 3,5, ]end{array}right; ; ; Rightarrow ; ; underline {; xin (-1,1)cup [, 3,4)cup (4,5, ]; }$