Chapter 7

$$ \log _{3}(7-x) \leq \frac{9}{16} \log _{2 \sqrt{2}}^{2} \frac{1}{4}+\log _{7-x} 9 $$

$$ \log _{3} x-\log _{3}^{2} x \leq \frac{3}{2} \log _{\frac{1}{2 \sqrt{2}}} 4 $$

$$ \left(\log _{5} x\right)^{2}+\left(\log _{5} x\right)<2 $$

$$ \frac{1-\log _{4} x}{1+\log _{2} x} \leq \frac{1}{2} $$

$$ \log _{x}(x-1) \geq 2 $$

$$ \log _{x}\left(x^{2}-\frac{3}{16}\right)>4 $$

$$ \log _{x}\left(16-6 x-x^{2}\right) \leq 1 $$

$$ \log _{x}\left(x^{3}-x^{2}-2 x\right)<3 $$

$$ \log _{x} \frac{(x+3)}{(x-1)}>1 $$

$$ \log _{x} \sqrt{21-4 x}>1 $$

$$ \log _{2 x}\left(x^{2}-5 x+6\right)<1 $$

$$ \log _{x+4}(5 x+20) \leq \log _{x+4}(x+4)^{2} $$

$$ \log _{2 x+3} x^{2}<1 $$

$$ \log _{2 x-x^{2}}\left(x-\frac{3}{2}\right)^{4}>0 $$

$$ \log _{\frac{x-1}{x+5}} \frac{3}{10}>0 $$

$$ \log _{\frac{1}{x}} \frac{2(x-2)}{(x+1)(x-5)} \geq 1 $$

$$ \log _{x+\frac{1}{x}}\left(x^{2}+\frac{1}{x^{2}}-4\right) \geq 1 $$

$$ \log _{\sqrt{x+1}-\sqrt{x-1}}\left(x^{2}-3 x+1\right) \geq 0 $$

$$ x^{\log ^{2} x-3 \log x+1}>1000 $$

$$ \left(\frac{x}{10}\right)^{\log x-2}<100 $$

$$ x^{\log x}>10 \cdot x^{-\log x}+3 $$

$$ \left(x^{2}-x-1\right)^{x^{2}-1}<1 $$

Solve the equation \(2 a(a-2) x=a-2\).

Solve the equation \((a-1) x^{2}+2(2 a+1) x+(4 a+3)=0\).

Solve the equation \(\frac{x^{2}+1}{a^{2} x-2 a}-\frac{1}{2-a x}=\frac{x}{a}\).

Solve the inequality \(\frac{7 x-11}{a+3}>(1+3 a) \frac{x}{4}\).

Solve the inequality \(a x^{2}-2 x+4>0\)

$$ x+\frac{1}{x} \geq 2 \text { if } x>0 $$

$$ x+\frac{1}{x} \leq-2 \text { if } x<0 $$

$$ \tan ^{2} x+\cot ^{2} x \geq 2 $$

$$ 10^{x}+10^{-x} \geq 2 $$

$$ |\cos x+\sec x| \geq 2 $$

$$ a^{2}+b^{2}+c^{2} \geq a b+b c+c a . $$

$$ (a b+x y)(a x+b y)>4 a b x y $$

$$ (b+c)(c+a)(a+b)>8 a b c $$

$$ (a+b+c)(b c+c a+a b)>9 a b c $$

$$ (a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)>9 $$

$$ \left(\frac{a}{e}+\frac{b}{f}+\frac{c}{g}\right)\left(\frac{e}{a}+\frac{f}{b}+\frac{g}{c}\right)>9 $$

$$ b^{2} c^{2}+c^{2} a^{2}+a^{2} b^{2}>a b c(a+b+c) . $$

$$ \frac{b c}{a}+\frac{c a}{b}+\frac{a b}{c} \geq a+b+c $$

$$ \frac{b c}{a^{3}}+\frac{c a}{b^{3}}+\frac{a b}{c^{3}} \geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c} $$

$$ \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq \frac{1}{\sqrt{b c}}+\frac{1}{\sqrt{c a}}+\frac{1}{\sqrt{a b}} $$

If \(a^{2}+b^{2}+c^{2}=1\) then show that \(-\frac{1}{2} \leq a b+b c+c a \leq 1\).

$$ 2\left(a^{3}+b^{3}+c^{3}\right) \geq b c(b+c)+c a(c+a)+a b(a+b) $$

$$ \frac{a^{3}+b^{3}+c^{3}}{3}>\left(\frac{a+b+c}{3}\right) \cdot\left(\frac{a^{2}+b^{2}+c^{2}}{3}\right) $$

If \(x_{i}>0, \quad i=1,2, \ldots n\), prove \(\left(x_{1}+x_{2}+\ldots x_{n}\right)\left(\frac{1}{x_{1}}+\frac{1}{x_{2}}+\ldots+\frac{1}{x_{n}}\right) \geq n^{2} .\)

If \(a+b+c=1\) then prove that \(\frac{8}{27 a b c}>\left(\frac{1}{a}-1\right)\left(\frac{1}{b}-1\right)\left(\frac{1}{c}-1\right)>8\).

If \(x+y+z=a\) then prove that \(8 x y z \leq(a-x)(a-y)(a-z) \leq \frac{8}{27} a^{3}\).

If \(a, b, c\) are positive real numbers such that \(a+b+c=1\), prove that \(\frac{b(1-b)}{a c}+\frac{c(1-c)}{a b}+\frac{a(1-a)}{b c} \geq 6\).

Prove that \(a b c d>81(s-a)(s-b)(s-c)(s-d)\) where \(3 s=a+b+c+d\).