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)
\({4^{|x + 1|}} > {4^2}\)
\( \Leftrightarrow |x + 1| > 2 \Leftrightarrow \left[ {\begin{array}{*{20}{c}}
{x + 1 > 2}\\
{x + 1 < - 2}
\end{array}} \right. \Leftrightarrow \left[ {\begin{array}{*{20}{c}}
{x > 1}\\
{x < - 3}
\end{array}} \right.\)
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c)
\({2^{ - {x^2} + 3x}} < {2^2}\)
\( \Leftrightarrow - {x^2} + 3x < 2 \)
\( \Leftrightarrow {x^2} - 3x + 2 > 0 \Leftrightarrow \left[ {\begin{array}{*{20}{c}}
{x < 1}\\
{x > 2}
\end{array}} \right.\)
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)
\(\eqalign{& {{{3^x}} \over {{3^x} - 2}} - 3 < 0 \Leftrightarrow {{ - {{2.3}^x} + 6} \over {{3^x} - 2}} < 0 \cr & \Leftrightarrow {{{3^x} - 3} \over {{3^x} - 2}} > 0 \Leftrightarrow \left[ {\matrix{{{3^x} > 3} \cr {{3^x} < 2} \cr} } \right. \Leftrightarrow \left[ {\matrix{{x > 1} \cr {x < {{\log }_3}2} \cr} } \right. \cr} \)