Lời giải:
Đặt biểu thức đã cho là $P$

\(2P=\frac{2}{\sqrt{1}+\sqrt{3}}+\frac{2}{\sqrt{5}+\sqrt{7}}+...+\frac{2}{\sqrt{97}+\sqrt{99}}>\frac{1}{\sqrt{1}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{9}}+....+\frac{1}{\sqrt{97}+\sqrt{99}}+\frac{1}{\sqrt{99}+\sqrt{101}}(*)\)

Mà:

\(\frac{1}{\sqrt{1}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+....+\frac{1}{\sqrt{97}+\sqrt{99}}+\frac{1}{\sqrt{99}+\sqrt{101}}\)
\(=\frac{\sqrt{3}-\sqrt{1}}{(\sqrt{1}+\sqrt{3})(\sqrt{3}-\sqrt{1})}+\frac{\sqrt{5}-\sqrt{3}}{(\sqrt{3}+\sqrt{5})(\sqrt{5}-\sqrt{3})}+....+\frac{\sqrt{101}-\sqrt{99}}{(\sqrt{99}+\sqrt{101})(\sqrt{101}-\sqrt{99})}\)

\(=\frac{\sqrt{3}-\sqrt{1}}{2}+\frac{\sqrt{5}-\sqrt{3}}{2}+...+\frac{\sqrt{101}-\sqrt{99}}{2}\)

\(=\frac{\sqrt{101}-\sqrt{1}}{2}>\frac{\sqrt{100}-1}{2}=\frac{9}{2}(**)\)

Từ \((*); (**)\Rightarrow 2P>\frac{9}{2}\Rightarrow P>\frac{9}{4}\) (đpcm)

Ta có :

\(2\left(\frac{1}{\sqrt{1}+\sqrt{3}}+\frac{1}{\sqrt{5}+\sqrt{7}}+...+\frac{1}{\sqrt{97}+\sqrt{99}}\right)\)

\(>\frac{1}{\sqrt{1}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{7}}+...+\frac{1}{\sqrt{97}+\sqrt{99}}+\frac{1}{\sqrt{99}+\sqrt{101}}\)

\(=\frac{1}{2}\left(\sqrt{3}-\sqrt{1}+\sqrt{5}-\sqrt{3}+...+\sqrt{101}-\sqrt{99}\right)\)

\(=\frac{1}{2}\left(\sqrt{101}-\sqrt{1}\right)>\frac{9}{2}\)

\(\Rightarrow\frac{1}{\sqrt{1}+\sqrt{3}}+\frac{1}{\sqrt{5}+\sqrt{7}}+...+\frac{1}{\sqrt{97}+\sqrt{99}}>\frac{9}{4}\left(đpcm\right)\)

Chúc bạn học tốt !!!

Đặt \(A=\frac{1}{\sqrt{1}+\sqrt{3}}+...+\frac{1}{\sqrt{97}+\sqrt{99}}\)

\(\Rightarrow2A=\frac{\left(\sqrt{3}\right)^2-\left(\sqrt{1}\right)^2}{\sqrt{1}+\sqrt{3}}+...+\frac{\left(\sqrt{99}\right)^2-\left(\sqrt{97}\right)^2}{\sqrt{97}+\sqrt{99}}\)

\(=\sqrt{3}-\sqrt{1}+\sqrt{5}-\sqrt{3}+...+\sqrt{99}-\sqrt{97}\)

\(=\sqrt{99}-1\)

\(\Rightarrow A=\frac{\sqrt{99}-1}{2}=\frac{2\sqrt{99}-2}{4}>\frac{9}{4}\left(đpcm\right)\)

Gọi \(A=\frac{1}{\sqrt{1}+\sqrt{3}}+\frac{1}{\sqrt{5}+\sqrt{7}}+...+\frac{1}{\sqrt{97}+\sqrt{99}}\)

\(\Rightarrow2A=\frac{2}{\sqrt{1}+\sqrt{3}}+\frac{2}{\sqrt{5}+\sqrt{7}}+...+\frac{2}{\sqrt{97}+\sqrt{99}}\)

\(=\frac{\left(\sqrt{3}\right)^2-\left(\sqrt{1}\right)^2}{\sqrt{3}+\sqrt{1}}+...+\frac{\left(\sqrt{99}\right)^2-\left(\sqrt{97}\right)^2}{\sqrt{99}+\sqrt{97}}\)

\(=\sqrt{3}-\sqrt{1}+\sqrt{5}-\sqrt{3}+...+\sqrt{99}-\sqrt{97}\)

\(=\sqrt{99}-1\)

Vậy \(A=\frac{\sqrt{99}-1}{2}=\frac{2\sqrt{99}-2}{4}>\frac{9}{4}\)

a) \(\dfrac{x+5}{5}+\dfrac{x+5}{7}+\dfrac{x+5}{9}=\dfrac{x+5}{11}+\dfrac{x+5}{13}\)

\(\Rightarrow\left(x+5\right)\left(\dfrac{1}{5}+\dfrac{1}{7}+\dfrac{1}{9}\right)=\left(x+5\right)\left(\dfrac{1}{11}+\dfrac{1}{13}\right)\)

\(\Rightarrow\dfrac{143}{315}\left(x+5\right)=\dfrac{24}{143}\left(x+5\right)\)

\(\Rightarrow\dfrac{143}{315}\left(x+5\right)-\dfrac{24}{143}\left(x+5\right)=0\)

\(\Rightarrow\left(x+5\right)\left(\dfrac{143}{315}-\dfrac{24}{143}\right)=0\)

\(\Rightarrow x+5=0\Rightarrow x=-5\)

b) \(\dfrac{x+2}{100}+\dfrac{x+3}{99}+\dfrac{x+4}{98}=\dfrac{x+5}{97}+\dfrac{x+6}{96}+\dfrac{x+7}{95}\)

\(\Rightarrow\)\(3+\dfrac{x+2}{100}+\dfrac{x+3}{99}+\dfrac{x+4}{98}=3+\dfrac{x+5}{97}+\dfrac{x+6}{96}+\dfrac{x+7}{95}\)

\(\Rightarrow\)\(1+\dfrac{x+2}{100}+1+\dfrac{x+3}{99}+1+\dfrac{x+4}{98}=1+\dfrac{x+5}{97}+1+\dfrac{x+6}{96}+1+\dfrac{x+7}{95}\)

\(\Rightarrow\)\(\dfrac{100}{100}+\dfrac{x+2}{100}+\dfrac{99}{99}+\dfrac{x+3}{99}+\dfrac{98}{98}+\dfrac{x+4}{98}=\dfrac{97}{97}+\dfrac{x+5}{97}+\dfrac{96}{96}+\dfrac{x+6}{96}+\dfrac{95}{95}+\dfrac{x+7}{95}\)\(\Rightarrow\)\(\dfrac{x+102}{100}+\dfrac{x+102}{99}+\dfrac{x+102}{98}=\dfrac{x+102}{97}+\dfrac{x+102}{96}+\dfrac{x+102}{95}\)

\(\Rightarrow\)\(\left(x+102\right)\left(\dfrac{1}{100}+\dfrac{1}{99}+\dfrac{1}{98}\right)=\left(x+102\right)\left(\dfrac{1}{97}+\dfrac{1}{96}+\dfrac{1}{95}\right)\)

\(\Rightarrow\)\(x+102=0\)

\(\Rightarrow x=-102\)

c) \(\left(x+2\right)-\left(x+3\right)>0\)

\(\Rightarrow x+2-x-3>0\Rightarrow-1>0\)

\(\Rightarrow x\in\varnothing\)

d) \(\left(x-5\right)\left(x+\dfrac{7}{3}\right)\ge0\)

TH1: \(\left\{{}\begin{matrix}x-5\ge0\\x+\dfrac{7}{3}\ge0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x\ge5\\x\ge\dfrac{-7}{3}\end{matrix}\right.\)

\(\Rightarrow x\ge\dfrac{-7}{3}\)

TH2: \(\left\{{}\begin{matrix}x-5\le0\\x+\dfrac{7}{3}\le0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x\le5\\x\le\dfrac{-7}{3}\end{matrix}\right.\)

\(\Rightarrow x\le5\)

TH3: \(\left[{}\begin{matrix}x-5=0\\x+\dfrac{7}{3}=0\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}x=5\\x=\dfrac{-7}{3}\end{matrix}\right.\)