A) \(\frac{7}{\left(x+3\right)\left(x+10\right)}+\frac{11}{\left(x+10\right)\left(x+21\right)}+\frac{13}{\left(x+21\right)\left(x+34\right)}\)

\(=\frac{\left(x+10\right)-\left(x+3\right)}{\left(x+3\right)\left(x+10\right)}+\frac{\left(x+21\right)-\left(x+10\right)}{\left(x+10\right)\left(x+21\right)}+\frac{\left(x+34\right)-\left(x+21\right)}{\left(x+21\right)\left(x+34\right)}\)

\(=\frac{1}{x+3}-\frac{1}{x+10}+\frac{1}{x+10}-\frac{1}{x+21}+\frac{1}{x+21}-\frac{1}{x+34}\)

\(=\frac{1}{x+3}-\frac{1}{x+34}\)

\(=\frac{\left(x+34\right)-\left(x+3\right)}{\left(x+3\right)\left(x+34\right)}\)\(=\frac{x}{\left(x+3\right)\left(x+34\right)}\)

\(\Rightarrow\left(x+34\right)-\left(x+3\right)=x\)

\(\Rightarrow x=31\)

Vậy, x = 31 

Bạn áp dụng: \(\frac{k}{x\cdot\left(x+k\right)}=\frac{1}{x}-\frac{1}{x+k}\) với    \(x,k\inℝ;x\ne0;x\ne-k\)

Chứng minh: \(\frac{1}{x}-\frac{1}{x+k}=\frac{x+k}{x\left(x+k\right)}-\frac{x}{x\left(x+k\right)}=\frac{x+k-x}{x\left(x+k\right)}=\frac{k}{x\left(x+k\right)}\)

B) \(\frac{\left(x-4\right)-\left(x-7\right)}{\left(x-7\right)\left(x-4\right)}+\frac{\left(x-7\right)-\left(x-13\right)}{\left(x-13\right)\left(x-7\right)}+\frac{\left(x-13\right)-\left(x-28\right)}{\left(x-28\right)\left(x-13\right)}\)

\(=\frac{1}{x-7}-\frac{1}{x-4}+\frac{1}{x-13}-\frac{1}{x-7}+\frac{1}{x-28}-\frac{1}{x-13}\)

\(=\frac{1}{x-28}-\frac{1}{x-4}=-\frac{5}{2}+\frac{1}{x-28}\)

\(\Leftrightarrow\frac{1}{x-28}-\frac{1}{x-4}-\frac{1}{x-28}=-\frac{5}{2}\)

\(\Leftrightarrow\frac{1}{x-4}=\frac{5}{2}\)

=> 5x - 20 = 2

=> 5x = 22 

\(\Rightarrow x=\frac{22}{5}=4,4\)

Vậy, x = 4,4

ghi sai để nên mik ko giải dc

a, \(\frac{1}{5.6}+\frac{1}{6.7}+...+\frac{1}{x\left(x+1\right)}=\frac{13}{90}\)

⇒ \(\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+...+\frac{1}{x}-\frac{1}{x+1}=\frac{13}{90}\)

⇒ \(\frac{1}{5}-\frac{1}{x+1}=\frac{13}{90}\)

⇒ \(\frac{1}{x+1}=\frac{1}{5}-\frac{13}{90}\)

⇒ \(\frac{1}{x+1}=\frac{18}{90}-\frac{13}{90}\)

⇒ \(\frac{1}{x+1}=\frac{1}{18}\)

⇒ x + 1 = 18

⇒ x = 17

Vậy x = 17

b, \(\frac{1}{1.4}+\frac{1}{4.7}+\frac{1}{7.10}+...+\frac{1}{x\left(x+3\right)}=\frac{49}{148}\)

⇒ \(\frac{3}{1.4}+\frac{3}{4.7}+\frac{3}{7.10}+...+\frac{3}{x\left(x+3\right)}=\frac{49.3}{148}\)

⇒ \(1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+...+\frac{1}{x}-\frac{1}{x+3}=\frac{147}{148}\)

⇒ \(1-\frac{1}{x+3}=\frac{147}{148}\)

⇒ \(\frac{1}{x+3}=1-\frac{147}{148}\)

⇒ \(\frac{1}{x+3}=\frac{1}{148}\)

⇒ x + 3 = 148

⇒ x = 145

Vậy x = 145

bài này khó ak nha

Vế trái: 4/(x+2).(x+6)+7/(x+6).(x+13)

<=>1/x+2 -1/x+6 +1/x+6 -1/x+13

<=>1/x+2-1/x+13

=> 1/x+2-1/x+13=2x+1/(x+2).(x+16) -3/(x+13).(x+16)

<=>1/x+2 - 1/x+13 + 1/x+13 - 1/x+16=2x+1/(x+2).(x+16)

<=>1/x+2 - 1/x+16=2x+1/(x+2).(x+16)

<=> 14/(x+2).(x+16)= 2x+1/(x+2).(x+16)

<=> 2x+1=14

<=> 2x=14-1

<=> 2x=13

<=> x=13:2

<=> x=13/2

Vậy x=13/2

Chắc là vầy. Mk cug ko chắc nữa

Vế trái: 4/(x+2).(x+6)+7/(x+6).(x+13)

<=>1/x+2 -1/x+6 +1/x+6 -1/x+13

<=>1/x+2-1/x+13

=> 1/x+2-1/x+13=2x+1/(x+2).(x+16) -3/(x+13).(x+16)

<=>1/x+2 - 1/x+13 + 1/x+13 - 1/x+16=2x+1/(x+2).(x+16)

<=>1/x+2 - 1/x+16=2x+1/(x+2).(x+16)

<=> 14/(x+2).(x+16)= 2x+1/(x+2).(x+16)

<=> 2x+1=14

<=> 2x=14-1

<=> 2x=13

<=> x=13:2

<=> x=13/2

Vậy x=13/2

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