Bài 8 (Trang 90 SGK Toán Giải Tích 12):

Giải các bất phương trình:

a) $2^{2x-1}+2^{2x-2}+2^{2x-3}\ge 448$;

b) ${(0,4)}^x-{(2,5)}^{x+1}>1,5$;

c) ${\log }_3\left[{\log }_{\frac{1}{2}}(x^2-1)\right]<1$;

d) ${{\log }_{0,2}}^{2}x-5.{\log }_{0,2}x<-6$.

Hướng dẫn giải:

a) $\frac{1}{2}.2^x + \frac{1}{4}.2^{2x} + \frac{1}{8}.2^{2x} \ge 448 \iff \frac{7}{8}.2^{2x} \ge 448 \iff 2^{2x} \ge 512 = 2^9 \iff 2x \ge 9 \iff x \ge \frac{9}{2}$

Vậy $\mathrm{S}=[\frac{9}{2};+\infty )$

b) Ta có: $0,4 = \frac{2}{5} \mathrm{và} 2,5 = \frac{5}{2}.$ 

      ${(0,4)}^{\mathrm{x}} - {(2,5)}^{\mathrm{x}+1} > 1,5 \iff {\left(\frac{2}{5}\right)}^{\mathrm{x}} - \frac{5}{2}.{\left(\frac{5}{2}\right)}^{\mathrm{x}} > \frac{3}{2}$

Đặt t= ${\left(\frac{2}{5}\right)}^x$ (t>0) ta có: $\mathrm{t} = {\left(\frac{2}{5}\right)}^{\mathrm{x}} (\mathrm{t} > 0), \mathrm{ta} \mathrm{có}: \mathrm{t}-\frac{5}{2}.\frac{1}{\mathrm{t}}>\frac{3}{2} \iff 2{\mathrm{t}}^2 - 3\mathrm{t} - 5 > 0 \iff [\begin{array}{l}\mathrm{t} < -1 \left(\mathrm{loại}\right)\\ \mathrm{t} > \frac{5}{2}\end{array}$

Với $\mathrm{t}>\frac{5}{2}\iff {\left(\frac{2}{5}\right)}^{\mathrm{x}}>{\left(\frac{2}{5}\right)}^{-1}\iff \mathrm{x} <-1$

Vậy $\mathrm{S}=(-\infty ;-1)$

c) $$\begin{gathered} {\log }_3\left({\log }_{\frac{1}{2}}\right(x^2-1\left)\right) < {\log }_{3}3 \iff 0<{\log }_{\frac{1}{2}}(x^2-1) <3 \\ \iff {\log }_{\frac{1}{2}}1 <{\log }_{\frac{1}{2}}(x^2-1) <{\log }_{\frac{1}{2}}\frac{1}{8} \iff 1 > x^2-1 > \frac{1}{8} \\ \iff 2 > x^2 > \frac{9}{8} \iff \frac{3}{2\sqrt{2}} < \left|x\right| < \sqrt{2} \iff [\begin{array}{l}\frac{3}{2\sqrt{2}} < x < \sqrt{2}\\ -\sqrt{2} < x < -\frac{3}{2\sqrt{2}}\end{array} \end{gathered}$$

Vậy $S=\left( -\sqrt{2};-\frac{3}{2\sqrt{2}}\right)\cup \left(\frac{3}{2\sqrt{2}};\sqrt{2}\right)$

d) Điều kiện x > 0. Đặt t = log_0,2x, ta có:

$$\begin{gathered} t^2 - 5t + 6 < 0 \iff 2 < t < 3 \iff {\log }_{0,2}{(0,2)}^2 < {\log }_{0,2}(0,2) \\ < {\log }_{0,2}{(0,2)}^3 \iff { (0,2)}^2 > x > (0,2{)}^3 \iff 0,008 < x <0,04 \end{gathered}$$

Vậy S=(0,008 ; 0,04).