Question

Giải các phương trình sau:

a) \(\frac{7}{x+2}=\frac{3}{x-5}\)

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

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

d) \(\frac{2x+1}{x-1}=\frac{5\left(x-1\right)}{x+1}\)

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

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

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

h) \(\frac{1}{7-x}=\frac{x-8}{x-7}-8\)

i)\(\frac{x+6}{x}=\frac{1}{2}+\frac{15}{2\left(x-6\right)}\)

Answer 1

a) ĐKXĐ : \(x\ne-2;x\ne5\)

\(\frac{7}{x+2}=\frac{3}{x-5}\)

<=> 3(x + 2) = 7(x - 5)

<=> 3x + 6 = 7x - 35

<=> 4x = 41

<=>x = 41/4 (tm)

Vậy x = 41/4 là ngiệm phương trình

b) ĐKXĐ \(x\ne\pm3\)

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

<=> \(\frac{\left(2x-1\right)\left(x-3\right)}{\left(x+3\right)\left(x-3\right)}=\frac{2x\left(x+3\right)}{\left(x-3\right)\left(x+3\right)}\)

<=> (2x - 1)(x - 3) = 2x(x + 3)

<=> 2x² - 7x + 3 = 2x² + 6x

<=> 13x = 3

<=> x = 3/13 (tm)

Vậy x = 3/13 là nghiệm phương trình

c) ĐKXĐ : \(x\ne-7;x\ne1,5\)

Khi đó \(\frac{3x-2}{x+7}=\frac{6x+1}{2x-3}\)

<=> \(\frac{\left(3x-2\right)\left(2x-3\right)}{\left(x+7\right)\left(2x-3\right)}=\frac{\left(6x+1\right)\left(x+7\right)}{\left(x+7\right)\left(2x-3\right)}\)

<=> (3x - 2)(2x - 3) = (6x + 1)(x + 7)

<=> 6x² - 13x + 6 = 6x² + 43x + 7

<=> 56x = -1

<=> x = -1/56 (tm) 

Vậy x = -1/56 là nghiệm phương trình

d) ĐKXĐ : \(x\ne\pm1\)

Khi đó \(\frac{2x+1}{x-1}=\frac{5\left(x-1\right)}{x+1}\)

<=> \(\frac{\left(2x+1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}=\frac{5\left(x-1\right)^2}{\left(x-1\right)\left(x+1\right)}\)

<=> (2x + 1)(x + 1) = 5(x - 1)²

<=> 2x² + 3x + 1 = 5x² - 10x + 5

<=> 3x² - 13x + 4 = 0

<=> 3x² - 12x - x + 4 = 0

<=> 3x(x - 4) - (x - 4) = 0

<=> (3x - 1)(x - 4) = 0

<=> \(\orbr{\begin{cases}3x-1=0\\x-4=0\end{cases}}\Leftrightarrow\orbr{\begin{cases}x=\frac{1}{3}\left(tm\right)\\x=4\left(tm\right)\end{cases}}\)

Vậy x \(\in\left\{\frac{1}{3};4\right\}\)là nghiệm phương trình

Answer 2

e) ĐKXĐ : \(x\ne1\)

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

<=> \(\frac{3x-5}{x-1}=2\)

<=> 3x - 5 = 2(x - 1) 

<=> 3x - 5 = 2x - 2

<=> x = 3 (tm) 

Vậy x = 3 là nghiệm phương trình

f) ĐKXĐ : \(x\ne-1\)

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

<=> \(\frac{3x+2}{x+1}=3\)

<=> 3x + 2 = 3(x + 1)

<=> 3x + 2 = 3x + 3

<=> 0x = 1

<=> \(x\in\varnothing\)

Vậy tập nghiệm phương trình S = \(\varnothing\)

g) ĐKXĐ : \(x\ne2\)

Khi đó \(\frac{1}{x-2}+3=\frac{x-3}{2-x}\)

<=>\(\frac{x-2}{x-2}=3\)

<=> (x - 2) = 3(x - 2)

<=> x - 2 = 3x - 6

<=> -2x = -4

<=> x = 2 (loại) 

Vậy tập nghiệm phương trình S = \(\varnothing\)

h) ĐKXĐ : \(x\ne7\)

Khi đó \(\frac{1}{7-x}=\frac{x-8}{x-7}-8\)

<=> \(\frac{x-7}{x-7}=8\)

<=> x - 7 = 8(x - 7)

<=> x - 7 = 8x - 56

<=> 7x = 49

<=> x = 7 (loại)

Vậy tập nghiệm phương trình S = \(\varnothing\)

i) ĐKXĐ : \(x\ne0;x\ne6\)

Ta có : \(\frac{x+6}{x}=\frac{1}{2}+\frac{15}{2\left(x-6\right)}\)

<=> \(\frac{x+6}{x}-\frac{15}{2\left(x-6\right)}=\frac{1}{2}\)

<=> \(\frac{2\left(x+6\right)\left(x-6\right)}{2x\left(x-6\right)}-\frac{15x}{2x\left(x-6\right)}=\frac{1}{2}\)

<=> \(\frac{2x^2-72-15x}{2x\left(x-6\right)}=\frac{1}{2}\)

<=> 4x² - 144 - 30x = 2x(x - 6) 

<=> 2x² - 18x - 144 = 0

<=> x² - 9x - 72 = 0

<=> x² - 9x + 81/4 - 72- 81/4 = 0

<=> \(\left(x-\frac{9}{2}\right)^2-\frac{369}{4}=0\)

<=> \(\left(x-\frac{9}{2}+\sqrt{\frac{369}{4}}\right)\left(x-\frac{9}{2}-\sqrt{\frac{369}{4}}\right)=0\)

<=> \(\orbr{\begin{cases}x=\frac{9}{2}-\sqrt{\frac{369}{4}}\\x=\frac{9}{2}+\sqrt{\frac{369}{4}}\end{cases}}\)(tm)

Vậy x \(\in\left\{\frac{9}{2}-\sqrt{\frac{369}{4}};\frac{9}{2}+\sqrt{\frac{369}{4}}\right\}\)

Answer 3

tự kết luận nhé ! 

e, \(\frac{4x-5}{x-1}=2+\frac{x}{x-1}\)ĐK : x \(\ne\)1

\(\Leftrightarrow\frac{4x-5}{x-1}=\frac{2x-2+x}{x-1}\Rightarrow4x+5=3x-2\Leftrightarrow x=-7\)

f, \(\frac{1-x}{x+1}+3=\frac{2x+3}{x+1}\)ĐK : x \(\ne\)-1

\(\Leftrightarrow\frac{1-x+3x+3}{x+1}=\frac{2x+3}{x+1}\Rightarrow4+2x=2x+3\Leftrightarrow0\ne-1\)

Vậy phương trình vô nghiệm 

g, \(\frac{1}{x-2}+3=\frac{x-3}{2-x}\)ĐK : x \(\ne\)2

\(\Leftrightarrow\frac{1+3x-6}{x-2}=\frac{-x+3}{x-2}\Rightarrow-5+3x=-x+3\Leftrightarrow4x=8\Leftrightarrow x=2\)

h, \(\frac{1}{7-x}=\frac{x-8}{x-7}-8\)ĐK : x \(\ne\)7

\(\Leftrightarrow\frac{-1}{x-7}=\frac{x-8-8x+56}{x-7}\Rightarrow-1=-7x+48\Leftrightarrow x=7\)

i, \(\frac{x+6}{x}=\frac{1}{2}+\frac{15}{2\left(x-6\right)}\)ĐK : x \(\ne\)0;6

\(\Leftrightarrow\frac{x+6}{x}=\frac{x-6+15}{2\left(x-6\right)}\Leftrightarrow\frac{x+6}{x}-\frac{x+9}{2\left(x-6\right)}=0\)

\(\Leftrightarrow\frac{2\left(x+6\right)\left(x-6\right)}{2x\left(x-6\right)}-\frac{x\left(x+9\right)}{2x\left(x-6\right)}=0\)

\(\Rightarrow2\left(x^2-36\right)-x^2-9x=0\Leftrightarrow2x^2-72-x^2-9x=0\)

\(\Leftrightarrow x^2-9x-72=0\)( check hộ bài mình làm ý '' i '' nhé, sai đâu để mình sửa ) 

\(\Leftrightarrow x=\frac{9\pm3\sqrt{41}}{2}\)