Advertisements (Quảng cáo)
Giải các bất phương trình mũ sau:
a) \({3^{|x – 2|}} < 9\)
b) \({4^{|x + 1|}} > 16\)
c) \({2^{ – {x^2} + 3x}} < 4\)
d) \({(\frac{7}{9})^{2{x^2} – 3x}} \ge \frac{9}{7}\)
e) \({11^{\sqrt {x + 6} }} \ge {11^x}\)
g) \({2^{2x – 1}} + {2^{2x – 2}} + {2^{2x – 3}} \ge 448\)
h)\({16^x} – {4^x} – 6 \le 0\)
i) \(\frac{{{3^x}}}{{{3^x} – 2}} < 3\)
Hướng dẫn làm bài:
a) \({3^{|x – 2|}} < {3^2}\)
\( \Leftrightarrow |x – 2| < 2\)
\( \Leftrightarrow – 2 < x – 2 < 2\)
\( \Leftrightarrow 0 < x < 4\)
b)
\({4^{|x + 1|}} > {4^2}\)
\( \Leftrightarrow |x + 1| > 2 \Leftrightarrow \left[ {\begin{array}{*{20}{c}}
{x + 1 > 2}\\
{x + 1 < – 2}
\end{array}} \right. \Leftrightarrow \left[ {\begin{array}{*{20}{c}}
{x > 1}\\
{x < – 3}
\end{array}} \right.\)
c)
Advertisements (Quảng cáo)
\({2^{ – {x^2} + 3x}} < {2^2}\)
\( \Leftrightarrow – {x^2} + 3x < 2 \)
\( \Leftrightarrow {x^2} – 3x + 2 > 0 \Leftrightarrow \left[ {\begin{array}{*{20}{c}}
{x < 1}\\
{x > 2}
\end{array}} \right.\)
d)
\({(\frac{7}{9})^{2{x^2} – 3x}} \ge {(\frac{7}{9})^{ – 1}}\)
\( \Leftrightarrow 2{x^2} – 3x \le – 1\)
\( \Leftrightarrow 2{x^2} – 3x + 1 \le 0 \Leftrightarrow \frac{1}{2} \le x \le 1\)
e)
\(\eqalign{& \sqrt {x + 6} \ge x \Leftrightarrow \left[ {\matrix{{\left\{ {\matrix{{x + 6 \ge 0} \cr {x < 0} \cr} } \right.} \cr {\left\{ {\matrix{{x \ge 0} \cr {x + 6 \ge {x^2}} \cr} } \right.} \cr} } \right. \cr & \Leftrightarrow \left[ {\matrix{{\left\{ {\matrix{{x \ge – 6} \cr {x < 0} \cr} } \right.} \cr {\left\{ {\matrix{{x \ge 0} \cr {{x^2} – x – 6 \le 0} \cr} } \right.} \cr} } \right. \Leftrightarrow \left[ {\matrix{{ – 6 \le x < 0} \cr {\left\{ {\matrix{{ – 2 \le x \le 3} \cr {x \ge 0} \cr} } \right.} \cr} } \right. \cr & \Leftrightarrow \left[ {\matrix{{ – 6 \le x < 0} \cr {0 \le x \le 3} \cr} } \right. \Leftrightarrow – 6 \le x \le 3 \cr}\)
g)
\(\frac{1}{2}{.2^{2x}} + \frac{1}{4}{.2^{2x}} + \frac{1}{8}{.2^{2x}} \ge 448\)
\( \Leftrightarrow {2^{2x}} \ge 512 \Leftrightarrow {2^{2x}} \ge {2^9} \Leftrightarrow x \ge \frac{9}{2}\)
h) Đặt t = 4x (t > 0), ta có hệ bất phương trình:
\(\eqalign{& \left\{ {\matrix{{{t^2} – t – 6 \le 0} \cr {t > 0} \cr} } \right. \Leftrightarrow \left\{ {\matrix{{ – 2 \le t \le 3} \cr {t > 0} \cr} } \right. \cr & \Leftrightarrow 0 < t \le 3 \Leftrightarrow 0 < {4^x} \le 3 \Leftrightarrow x \le {\log _4}3 \cr} \)
i)
\(\eqalign{& {{{3^x}} \over {{3^x} – 2}} – 3 < 0 \Leftrightarrow {{ – {{2.3}^x} + 6} \over {{3^x} – 2}} < 0 \cr & \Leftrightarrow {{{3^x} – 3} \over {{3^x} – 2}} > 0 \Leftrightarrow \left[ {\matrix{{{3^x} > 3} \cr {{3^x} < 2} \cr} } \right. \Leftrightarrow \left[ {\matrix{{x > 1} \cr {x < {{\log }_3}2} \cr} } \right. \cr} \)