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Bài 81. Giải bất phương trình:
\(\eqalign{
& a)\,{\log _5}\left( {3x – 1} \right) < 1\,; \cr
& c)\,{\log _{0,5}}\left( {{x^2} – 5x + 6} \right) \ge – 1\,; \cr} \)
\(\eqalign{
& b)\,{\log _{{1 \over 3}}}\left( {5x – 1} \right) > 0\,; \cr
& d)\,{\log _3}{{1 – 2x} \over x} \le 0. \cr} \)
\(\eqalign{
& a)\,{\log _5}\left( {3x – 1} \right) < 1 \Leftrightarrow {\log _5}\left( {3x – 1} \right) < {\log _5}5 \cr
& \Leftrightarrow 0 < 3x – 1 < 5 \Leftrightarrow {1 \over 3} < x < 2 \cr} \)
Vậy \(S = \left( {{1 \over 3};2} \right)\)
\(\eqalign{
& b)\,{\log _{{1 \over 3}}}\left( {5x – 1} \right) > 0 \Leftrightarrow {\log _{{1 \over 3}}}\left( {5x – 1} \right) > {\log _{{1 \over 3}}}1 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Leftrightarrow 0 < 5x – 1 < 1 \Leftrightarrow {1 \over 5} < x < {2 \over 5} \cr} \)
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Vậy \(S = \left( {{1 \over 5};{2 \over 5}} \right)\)
\(\eqalign{
& c)\,{\log _{0,5}}\left( {{x^2} – 5x + 6} \right) \ge – 1 \Leftrightarrow \,{\log _{0,5}}\left( {{x^2} – 5x + 6} \right) \ge {\log _{0,5}}2 \cr
& \Leftrightarrow 0 < {x^2} – 5x + 6 \le 2 \Leftrightarrow \left\{ \matrix{
{x^2} – 5x + 6 > 0 \hfill \cr
{x^2} – 5x + 4 \le 0 \hfill \cr} \right. \cr
& \Leftrightarrow \left\{ \matrix{
x < 2\,\text { hoặc }\,x > 3 \hfill \cr
1 \le x \le 4 \hfill \cr} \right. \Leftrightarrow 1 \le x < 2\,\,\text { hoặc }\,\,3 < x \le 4 \cr} \)
Vậy tập nghiệm của bất phương trình là: \(S = \left[ {1;2} \right) \cup \left( {3;4} \right]\)
\(\eqalign{
& d)\,{\log _3}{{1 – 2x} \over x} \le 0 \Leftrightarrow {\log _3}{{1 – 2x} \over x} \le {\log _3}1 \cr
& \Leftrightarrow 0 < {{1 – 2x} \over x} \le 1 \Leftrightarrow \left\{ \matrix{
{{1 – 2x} \over x} > 0 \hfill \cr
{{1 – 2x} \over x} – 1 \le 0 \hfill \cr} \right. \cr
& \Leftrightarrow \left\{ \matrix{
0 < x < {1 \over 2} \hfill \cr
{{1 – 3x} \over x} \le 0 \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
0 < x < {1 \over 2} \hfill \cr
x \le 0\,\text { hoặc }\,x \ge {1 \over 3} \hfill \cr} \right. \cr
& \Leftrightarrow {1 \over 3} \le x < {1 \over 2} \cr} \)
Vậy \(S = \left[ {{1 \over 3};{1 \over 2}} \right)\)