a) \(\frac{x + 2004}{x + 2005} + \frac{x + 2005}{2006} < \frac{x + 2006}{2007} + \frac{x + 2007}{2008} \Rightarrow \left(\right. \frac{x + 2004}{2005} - 1 \left.\right) + \left(\right. \frac{x + 2005}{2006} - 1 \left.\right) < \left(\right. \frac{x + 2006}{2007} - 1 \left.\right) + \left(\right. \frac{x + 2007}{2008} - 1 \left.\right) \Rightarrow \frac{x - 1}{2005} + \frac{x - 1}{2006} < \frac{x - 1}{2007} + \frac{x - 1}{2008} \Rightarrow \frac{x - 1}{2005} + \frac{x - 1}{2006} - \frac{x - 1}{2007} - \frac{x - 1}{2008} < 0\)

\(\Rightarrow \left(\right. x - 1 \left.\right) \left(\right. \frac{1}{2005} + \frac{1}{2006} - \frac{1}{2007} - \frac{1}{2008} \left.\right) < 0 \left(\right. a \left.\right)\)

Nhận thấy: \(\frac{1}{2005} > \frac{1}{2007} , \frac{1}{2006} > \frac{1}{2008} \Rightarrow \frac{1}{2005} - \frac{1}{2007} > 0 , \frac{1}{2006} - \frac{1}{2008} > 0 \Rightarrow \frac{1}{2005} + \frac{1}{2006} - \frac{1}{2007} - \frac{1}{2008} > 0\)

\(\left(\right. a \left.\right) \Rightarrow x - 1 < 0 \Leftrightarrow x < 1\)

Vậy bpt co nghiem x<1

b) \(\frac{x - 2}{2002} + \frac{x - 4}{2000} < \frac{x - 3}{2001} + \frac{x - 5}{1999} \Rightarrow \left(\right. \frac{x - 2}{2002} - 1 \left.\right) + \left(\right. \frac{x - 4}{2000} - 1 \left.\right) < \left(\right. \frac{x - 3}{2001} - 1 \left.\right) + \left(\right. \frac{x - 5}{1999} - 1 \left.\right) \Rightarrow \frac{x - 2004}{2002} + \frac{x - 2004}{2000} < \frac{x - 2004}{2001} + \frac{x - 2004}{1999} \Rightarrow \frac{x - 2004}{2002} + \frac{x - 2004}{2000} - \frac{x - 2004}{2001} - \frac{x - 2004}{1999} < 0\)

\(\Rightarrow \left(\right. x - 2004 \left.\right) \left(\right. \frac{1}{2002} + \frac{1}{2000} - \frac{1}{2001} - \frac{1}{1999} \left.\right) < 0 \left(\right. b \left.\right)\)

Nhận thấy: \(\frac{1}{2002} < \frac{1}{2001} , \frac{1}{2000} < \frac{1}{1999} \Rightarrow \frac{1}{2002} - \frac{1}{2001} < 0 , \frac{1}{2000} - \frac{1}{1999} < 0 \Rightarrow \frac{1}{2002} + \frac{1}{2000} - \frac{1}{2001} - \frac{1}{1999} < 0\)

\(\left(\right. b \left.\right) \Rightarrow x - 2004 > 0 \Leftrightarrow x > 2004\)

Vay bpt co nghiem x>2024

c) \(\frac{x - a b}{a + b} + \frac{x - b c}{b + c} + \frac{x - a c}{a + c} > a + b + c\)

\(\frac{x - a b}{a + b} - c + \frac{x - b c}{b + c} - a + \frac{x - a c}{a + c} - b > 0\)

\(\frac{x - a b - a c - b c}{a + b} + \frac{x - b c - a b - a c}{b + c} + \frac{x - a c - b c - a b}{a + c} > 0\)

\(\left(\right. x - a b - a c - b c \left.\right) \left(\right. \frac{1}{a + b} + \frac{1}{b + c} + \frac{1}{a + c} \left.\right) > 0\)

\(x - a b - a c - b c > 0\) do \(a,b;c>0\Rightarrow\frac{1}{a + b}+\frac{1}{b + c}+\frac{1}{a + c}>0\)

\(x > a b + a c + b c\)

Vay bpt co nghiem x<ab+ac+bc\(\)

a) \(\frac{x + 2004}{x + 2005} + \frac{x + 2005}{2006} < \frac{x + 2006}{2007} + \frac{x + 2007}{2008} \Rightarrow \left(\right. \frac{x + 2004}{2005} - 1 \left.\right) + \left(\right. \frac{x + 2005}{2006} - 1 \left.\right) < \left(\right. \frac{x + 2006}{2007} - 1 \left.\right) + \left(\right. \frac{x + 2007}{2008} - 1 \left.\right) \Rightarrow \frac{x - 1}{2005} + \frac{x - 1}{2006} < \frac{x - 1}{2007} + \frac{x - 1}{2008} \Rightarrow \frac{x - 1}{2005} + \frac{x - 1}{2006} - \frac{x - 1}{2007} - \frac{x - 1}{2008} < 0\)

\(\Rightarrow \left(\right. x - 1 \left.\right) \left(\right. \frac{1}{2005} + \frac{1}{2006} - \frac{1}{2007} - \frac{1}{2008} \left.\right) < 0 \left(\right. a \left.\right)\)

Nhận thấy: \(\frac{1}{2005} > \frac{1}{2007} , \frac{1}{2006} > \frac{1}{2008} \Rightarrow \frac{1}{2005} - \frac{1}{2007} > 0 , \frac{1}{2006} - \frac{1}{2008} > 0 \Rightarrow \frac{1}{2005} + \frac{1}{2006} - \frac{1}{2007} - \frac{1}{2008} > 0\)

\(\left(\right. a \left.\right) \Rightarrow x - 1 < 0 \Leftrightarrow x < 1\)

Vậy bpt co nghiem x<1

b) \(\frac{x - 2}{2002} + \frac{x - 4}{2000} < \frac{x - 3}{2001} + \frac{x - 5}{1999} \Rightarrow \left(\right. \frac{x - 2}{2002} - 1 \left.\right) + \left(\right. \frac{x - 4}{2000} - 1 \left.\right) < \left(\right. \frac{x - 3}{2001} - 1 \left.\right) + \left(\right. \frac{x - 5}{1999} - 1 \left.\right) \Rightarrow \frac{x - 2004}{2002} + \frac{x - 2004}{2000} < \frac{x - 2004}{2001} + \frac{x - 2004}{1999} \Rightarrow \frac{x - 2004}{2002} + \frac{x - 2004}{2000} - \frac{x - 2004}{2001} - \frac{x - 2004}{1999} < 0\)

\(\Rightarrow \left(\right. x - 2004 \left.\right) \left(\right. \frac{1}{2002} + \frac{1}{2000} - \frac{1}{2001} - \frac{1}{1999} \left.\right) < 0 \left(\right. b \left.\right)\)

Nhận thấy: \(\frac{1}{2002} < \frac{1}{2001} , \frac{1}{2000} < \frac{1}{1999} \Rightarrow \frac{1}{2002} - \frac{1}{2001} < 0 , \frac{1}{2000} - \frac{1}{1999} < 0 \Rightarrow \frac{1}{2002} + \frac{1}{2000} - \frac{1}{2001} - \frac{1}{1999} < 0\)

\(\left(\right. b \left.\right) \Rightarrow x - 2004 > 0 \Leftrightarrow x > 2004\)

Vay bpt co nghiem x>2024

c) \(\frac{x - a b}{a + b} + \frac{x - b c}{b + c} + \frac{x - a c}{a + c} > a + b + c\)

\(\frac{x - a b}{a + b} - c + \frac{x - b c}{b + c} - a + \frac{x - a c}{a + c} - b > 0\)

\(\frac{x - a b - a c - b c}{a + b} + \frac{x - b c - a b - a c}{b + c} + \frac{x - a c - b c - a b}{a + c} > 0\)

\(\left(\right. x - a b - a c - b c \left.\right) \left(\right. \frac{1}{a + b} + \frac{1}{b + c} + \frac{1}{a + c} \left.\right) > 0\)

\(x - a b - a c - b c > 0\) do \(a,b;c>0\Rightarrow\frac{1}{a + b}+\frac{1}{b + c}+\frac{1}{a + c}>0\)

\(x > a b + a c + b c\)

Vay bpt co nghiem x<ab+ac+bc\(\)

a) \(\frac{x + 2004}{x + 2005} + \frac{x + 2005}{2006} < \frac{x + 2006}{2007} + \frac{x + 2007}{2008} \Rightarrow \left(\right. \frac{x + 2004}{2005} - 1 \left.\right) + \left(\right. \frac{x + 2005}{2006} - 1 \left.\right) < \left(\right. \frac{x + 2006}{2007} - 1 \left.\right) + \left(\right. \frac{x + 2007}{2008} - 1 \left.\right) \Rightarrow \frac{x - 1}{2005} + \frac{x - 1}{2006} < \frac{x - 1}{2007} + \frac{x - 1}{2008} \Rightarrow \frac{x - 1}{2005} + \frac{x - 1}{2006} - \frac{x - 1}{2007} - \frac{x - 1}{2008} < 0\)

\(\Rightarrow \left(\right. x - 1 \left.\right) \left(\right. \frac{1}{2005} + \frac{1}{2006} - \frac{1}{2007} - \frac{1}{2008} \left.\right) < 0 \left(\right. a \left.\right)\)

Nhận thấy: \(\frac{1}{2005} > \frac{1}{2007} , \frac{1}{2006} > \frac{1}{2008} \Rightarrow \frac{1}{2005} - \frac{1}{2007} > 0 , \frac{1}{2006} - \frac{1}{2008} > 0 \Rightarrow \frac{1}{2005} + \frac{1}{2006} - \frac{1}{2007} - \frac{1}{2008} > 0\)

\(\left(\right. a \left.\right) \Rightarrow x - 1 < 0 \Leftrightarrow x < 1\)

Vậy bpt co nghiem x<1

b) \(\frac{x - 2}{2002} + \frac{x - 4}{2000} < \frac{x - 3}{2001} + \frac{x - 5}{1999} \Rightarrow \left(\right. \frac{x - 2}{2002} - 1 \left.\right) + \left(\right. \frac{x - 4}{2000} - 1 \left.\right) < \left(\right. \frac{x - 3}{2001} - 1 \left.\right) + \left(\right. \frac{x - 5}{1999} - 1 \left.\right) \Rightarrow \frac{x - 2004}{2002} + \frac{x - 2004}{2000} < \frac{x - 2004}{2001} + \frac{x - 2004}{1999} \Rightarrow \frac{x - 2004}{2002} + \frac{x - 2004}{2000} - \frac{x - 2004}{2001} - \frac{x - 2004}{1999} < 0\)

\(\Rightarrow \left(\right. x - 2004 \left.\right) \left(\right. \frac{1}{2002} + \frac{1}{2000} - \frac{1}{2001} - \frac{1}{1999} \left.\right) < 0 \left(\right. b \left.\right)\)

Nhận thấy: \(\frac{1}{2002} < \frac{1}{2001} , \frac{1}{2000} < \frac{1}{1999} \Rightarrow \frac{1}{2002} - \frac{1}{2001} < 0 , \frac{1}{2000} - \frac{1}{1999} < 0 \Rightarrow \frac{1}{2002} + \frac{1}{2000} - \frac{1}{2001} - \frac{1}{1999} < 0\)

\(\left(\right. b \left.\right) \Rightarrow x - 2004 > 0 \Leftrightarrow x > 2004\)

Vay bpt co nghiem x>2024

c) \(\frac{x - a b}{a + b} + \frac{x - b c}{b + c} + \frac{x - a c}{a + c} > a + b + c\)

\(\frac{x - a b}{a + b} - c + \frac{x - b c}{b + c} - a + \frac{x - a c}{a + c} - b > 0\)

\(\frac{x - a b - a c - b c}{a + b} + \frac{x - b c - a b - a c}{b + c} + \frac{x - a c - b c - a b}{a + c} > 0\)

\(\left(\right. x - a b - a c - b c \left.\right) \left(\right. \frac{1}{a + b} + \frac{1}{b + c} + \frac{1}{a + c} \left.\right) > 0\)

\(x - a b - a c - b c > 0\) do \(a,b;c>0\Rightarrow\frac{1}{a + b}+\frac{1}{b + c}+\frac{1}{a + c}>0\)

\(x > a b + a c + b c\)

Vay bpt co nghiem x<ab+ac+bc\(\)

a) \(\frac{x + 2004}{x + 2005} + \frac{x + 2005}{2006} < \frac{x + 2006}{2007} + \frac{x + 2007}{2008} \Rightarrow \left(\right. \frac{x + 2004}{2005} - 1 \left.\right) + \left(\right. \frac{x + 2005}{2006} - 1 \left.\right) < \left(\right. \frac{x + 2006}{2007} - 1 \left.\right) + \left(\right. \frac{x + 2007}{2008} - 1 \left.\right) \Rightarrow \frac{x - 1}{2005} + \frac{x - 1}{2006} < \frac{x - 1}{2007} + \frac{x - 1}{2008} \Rightarrow \frac{x - 1}{2005} + \frac{x - 1}{2006} - \frac{x - 1}{2007} - \frac{x - 1}{2008} < 0\)

\(\Rightarrow \left(\right. x - 1 \left.\right) \left(\right. \frac{1}{2005} + \frac{1}{2006} - \frac{1}{2007} - \frac{1}{2008} \left.\right) < 0 \left(\right. a \left.\right)\)

Nhận thấy: \(\frac{1}{2005} > \frac{1}{2007} , \frac{1}{2006} > \frac{1}{2008} \Rightarrow \frac{1}{2005} - \frac{1}{2007} > 0 , \frac{1}{2006} - \frac{1}{2008} > 0 \Rightarrow \frac{1}{2005} + \frac{1}{2006} - \frac{1}{2007} - \frac{1}{2008} > 0\)

\(\left(\right. a \left.\right) \Rightarrow x - 1 < 0 \Leftrightarrow x < 1\)

Vậy bpt co nghiem x<1

b) \(\frac{x - 2}{2002} + \frac{x - 4}{2000} < \frac{x - 3}{2001} + \frac{x - 5}{1999} \Rightarrow \left(\right. \frac{x - 2}{2002} - 1 \left.\right) + \left(\right. \frac{x - 4}{2000} - 1 \left.\right) < \left(\right. \frac{x - 3}{2001} - 1 \left.\right) + \left(\right. \frac{x - 5}{1999} - 1 \left.\right) \Rightarrow \frac{x - 2004}{2002} + \frac{x - 2004}{2000} < \frac{x - 2004}{2001} + \frac{x - 2004}{1999} \Rightarrow \frac{x - 2004}{2002} + \frac{x - 2004}{2000} - \frac{x - 2004}{2001} - \frac{x - 2004}{1999} < 0\)

\(\Rightarrow \left(\right. x - 2004 \left.\right) \left(\right. \frac{1}{2002} + \frac{1}{2000} - \frac{1}{2001} - \frac{1}{1999} \left.\right) < 0 \left(\right. b \left.\right)\)

Nhận thấy: \(\frac{1}{2002} < \frac{1}{2001} , \frac{1}{2000} < \frac{1}{1999} \Rightarrow \frac{1}{2002} - \frac{1}{2001} < 0 , \frac{1}{2000} - \frac{1}{1999} < 0 \Rightarrow \frac{1}{2002} + \frac{1}{2000} - \frac{1}{2001} - \frac{1}{1999} < 0\)

\(\left(\right. b \left.\right) \Rightarrow x - 2004 > 0 \Leftrightarrow x > 2004\)

Vay bpt co nghiem x>2024

c) \(\frac{x - a b}{a + b} + \frac{x - b c}{b + c} + \frac{x - a c}{a + c} > a + b + c\)

\(\frac{x - a b}{a + b} - c + \frac{x - b c}{b + c} - a + \frac{x - a c}{a + c} - b > 0\)

\(\frac{x - a b - a c - b c}{a + b} + \frac{x - b c - a b - a c}{b + c} + \frac{x - a c - b c - a b}{a + c} > 0\)

\(\left(\right. x - a b - a c - b c \left.\right) \left(\right. \frac{1}{a + b} + \frac{1}{b + c} + \frac{1}{a + c} \left.\right) > 0\)

\(x - a b - a c - b c > 0\) do \(a,b;c>0\Rightarrow\frac{1}{a + b}+\frac{1}{b + c}+\frac{1}{a + c}>0\)

\(x > a b + a c + b c\)

Vay bpt co nghiem x<ab+ac+bc\(\)