Question

Решите неравенство log3 (x-1)<=2 , log0.2 (2-x)>-1

Answer 1

$1) log_3(x-1) \leq 2 \\ 2) log_{0.2}(2-x)\ \textgreater \ -1 \\ 1) x-1\ \textgreater \ 0 \iff x\ \textgreater \ 1 \\ x-1 \leq 3^2 \iff x \leq 1+9 \iff x \leq 10 \\ x \in (1;10] \\ 2) 2-x\ \textgreater \ 0 \iff -x\ \textgreater \ -2 \iff x\ \textless \ 2 \\ 2-x\ \textless \ 0.2^{-1} \iff -x\ \textless \ -2+5 \iff -x\ \textless \ 3 \iff x\ \textgreater \ -3 \\ x \in(-3;2)$

Answer 2

1) x-1>0⇒x>1
x-1≤3²
x-1≤9
x≤10
$\left \{ {{x\ \textgreater \ 1} \atop {x \leq 10}} \right.$⇒x∈(1;10I
2)2-x>0⇒x<2
2-x<(0,2)⁻¹
2-x<5
2-5<x
x>-3
$\left \{ {{x\ \textless \ 2} \atop {x\ \textgreater \ -3}} \right.$⇒
x∈(-3;2)