chứng minh các BĐT   

1.\(\frac{a+c}{a+b}+\frac{b+d}{b+c}+\frac{c+a}{c+d}+\frac{b+d}{d+a}\ge4\)với a,b,c,d >0

2.\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\ge4\left(\frac{1}{2a+b+c}+\frac{1}{2b+c+d}+\frac{1}{2c+d+a}+\frac{1}{2d+a+b}\right)\)

3.\(\frac{1}{a^4+b^4+c^4}+\frac{2}{a^2b^2+b^2c^2+c^2a^2}\ge\left(\frac{3}{a^2+b^2+c^2}\right)^2\\ \)với a,b,c>0

4.\(\frac{1}{3x-2}-\frac{1}{x-10}+\frac{1}{13-2x}\ge\frac{3}{7}\)vói x,y t/m\(\frac{2}{3}< x< \frac{13}{2}\)

Giải các PT sau :

a,\(\frac{1-6x}{x-2}+\frac{9x+4}{x+2}=\frac{x\left(3x-2\right)+1}{\left(x+2\right)\left(x-2\right)}\)

b,(2 - 3x) (x + 11) = (3x - 2) (2 - 5x)

c,\(\frac{3x-2}{6}-5=\frac{3-2\left(x+7\right)}{4}\)

d,\(\frac{x}{x-3}+\frac{x}{x +1}=\frac{2x}{\left(x+1\right)\left(x-3\right)}\)

e,\(\frac{x+1}{5}+\frac{x+2}{4}=\frac{x+3}{3}+\frac{x+4}{2}\)

a,\(\frac{3}{x}+\frac{1}{x+3}+\frac{3}{x+6}+\frac{1}{x+7}=\frac{1}{1-x}\)

b, \(\frac{1}{x-5}+\frac{1}{x-2}+\frac{1}{x-1}+\frac{1}{x}+\frac{1}{x+3}=\frac{3x-3}{4}\)

c,\(\frac{1}{x-3}+\frac{1}{3x+1}+\frac{10x-13}{4x-6}=\frac{1}{x+1}+\frac{1}{2x-1}+\frac{1}{3x+7}\)

d,\(\frac{x^2+x+1}{2x-1}\left(\frac{3x^2-x+5}{4x-2}-3\right)=8\)

e,\(\frac{2x^2-3}{3x-1}\left(2x-\frac{7+4x}{3x-1}\right)=2\)

f,\(\frac{x\left(3x-1\right)\left(3x^2+1\right)\left(6x^2-3x-1\right)}{\left(x+1\right)^3}=\frac{1}{2}\)

g, \(x\left(x^2+2\right)\left(x^2+2x+8+\frac{12}{x-2}\right)=3\left(x-2\right)\)

6,Thực hiện phép tính

1,\(\frac{2a^3-2b^3}{3a+3b}.\frac{6a+6b}{a^2-2ab+b^2}\)

2,\(\frac{a^2+ab}{b-a}:\frac{a+b}{2a^2-2b^2}\)

3,\(\frac{x+y}{y-x}:\frac{x^2+xy}{3x^2-3y^2}\)

4,\(\frac{1-4x^2}{x^2+4x}:\frac{2-4x}{3x}\)

5,\(\frac{5x-15}{4x+4}:\frac{x^2-9}{x^2+2x+1}\)

6,\(\frac{6x+48}{7x-7}:\frac{x^2-64}{x^2-2x+1}\)

Thực hiện phép tính :

a)\(\frac{x^2}{\left(x-y\right)^2\left(x+y\right)}-\frac{2xy^2}{x^4-2x^2y^2+y^4}+\frac{y^2}{\left(x^2-y^2\right)\left(x+y\right)}\)  

b)\(\frac{1}{x-1}-\frac{1}{x+1}-\frac{2}{x^2+1}-\frac{4}{x^4+1}-\frac{8}{x^{8+1}}-\frac{16}{x^{16}+1}\)

c)\(\frac{1}{x^2+6x+9}+\frac{1}{6x-x^2-9}+\frac{x}{x^2-9}\)

d)\(\frac{a}{x^2+ax}+\frac{a}{x^2+3ax+2a^2}+\frac{a}{x^2+5ax+6a^2}+....+\frac{a}{x^2+19ax+90a^2}+\frac{1}{x+10a}\)

bài 1:giải các pt sau:

a/\(\frac{1-x}{x+1}\)+3=\(\frac{2x+3}{x+1}\)

b/\(\frac{\left(x+2\right)^2}{2x-3}-1=\frac{x^2+10}{2x-3}\)

c/\(\frac{1}{x+1}-\frac{5}{x-2}=\frac{15}{\left(x+1\right)\left(2-x\right)}\)

d/\(\frac{1-6x}{x-2}+\frac{9x+4}{x+2}=\frac{x\left(3x-2\right)+1}{x^2-4}\)

e/\(\frac{12}{1-9x^2}=\frac{1-3x}{1+3x}-\frac{1+3x}{1-3x}\)

f\(\frac{x+4}{x^2-3x+2}+\frac{x+1}{x^2-4x+3}=\frac{2x+5}{x^2-4x+3}\)

Giải các  phương trình sau :

\(a,6x^2-5x+3=2x-3x\left(3-2x\right)\)

\(b,\frac{2\left(x-4\right)}{4}-\frac{3+2x}{10}=x+\frac{1-x}{5}\)

\(c,\frac{2x}{3}+\frac{3x-5}{4}=\frac{3\left(2x-1\right)}{2}-\frac{7}{6}\)

\(d,\frac{6x+5}{2}-\frac{10x+3}{4}=2x+\frac{2x+1}{2}\)

\(e,\left(x-4\right)\left(x+4\right)-2\left(3x-2\right)=\left(x-4\right)^2\)

a) \(\left(\frac{1}{x^2+x}-\frac{2-x}{x+1}\right):\left(\frac{1}{x}+x-2\right)\)

b) \(\left(\frac{3x}{1-3x}+\frac{2x}{3x+1}\right):\frac{6x^2+10x}{1-6x+9x^2}\)

c) \(\frac{x^3}{x-1}-\frac{x^2}{x+1}-\frac{1}{x-1}+\frac{1}{x+1}\)

d) \(\frac{x^3+x^2-2x-20}{x^2-4}-\frac{5}{x+2}+\frac{3}{x-2}\)

e) \(\left[\frac{x^2-y^2}{xy}-\frac{1}{x+y}\left(\frac{x^2}{y}-\frac{y^2}{x}\right)\right]:\frac{x-y}{x}\)