Question

Thực hiện phép trừ:

1. 1/x-1 - 1/x+1

2. x/ x^2-1 - 1/x-1

3. 1/y.(x-y) - 1/ x ( x-y )

4. 1/x - 1/ x-1

5. 1/x - 1/ x+1

6. 1/ 2x^2 - 10x - 1/ x-5

7. x-1/x^2 -1 . x+1/x+3

8. 2/ 2x^2 +10x . x+5/3x

Answer 1

1. \(\dfrac{1}{x-1}-\dfrac{1}{x+1}\)

\(=\dfrac{1.\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}-\dfrac{1\left(x-1\right)}{\left(x+1\right)\left(x-1\right)}\)

\(=\dfrac{x+1}{\left(x+1\right)\left(x-1\right)}-\dfrac{x-1}{\left(x+1\right)\left(x-1\right)}\)

\(=\dfrac{x+1+\left(-x+1\right)}{\left(x+1\right)\left(x-1\right)}\)

\(=\dfrac{x+1-x+1}{\left(x+1\right)\left(x-1\right)}=\dfrac{1}{x^2-1}\)

2. \(\dfrac{x}{x^2-1}-\dfrac{1}{x-1}\)

\(=\dfrac{x}{\left(x+1\right)\left(x-1\right)}-\dfrac{x+1}{\left(x+1\right)\left(x-1\right)}\)

\(=\dfrac{x}{\left(x+1\right)\left(x-1\right)}+\dfrac{-\left(x+1\right)}{\left(x+1\right)\left(x-1\right)}\)

\(=\dfrac{x+\left(-x-1\right)}{\left(x+1\right)\left(x-1\right)}=\dfrac{-1}{x^2-1}\)

3. \(\dfrac{1}{x\left(x-y\right)}-\dfrac{1}{x\left(x-y\right)}\)

\(=\dfrac{1}{y\left(x-y\right)}+\dfrac{-1}{x\left(x-y\right)}\)

\(=\dfrac{1x}{y\left(x-y\right)x}+\dfrac{-1y}{x\left(x-y\right)y}\)

\(=\dfrac{x}{xy\left(x-y\right)}+\dfrac{-y}{xy\left(x-y\right)}\)

\(=\dfrac{x-y}{xy\left(x-y\right)}=\dfrac{1}{xy}\)

4. \(\dfrac{1}{x}-\dfrac{1}{x-1}\)

\(=\dfrac{1\left(x-1\right)}{x\left(x-1\right)}-\dfrac{1x}{\left(x-1\right)x}\)

\(=\dfrac{x-1}{x\left(x-1\right)}+\dfrac{-x}{x\left(x-1\right)}\)

\(=\dfrac{\left(x-1\right)-x}{x\left(x-1\right)}\)

\(=\dfrac{-1}{x\left(x-1\right)}\)

5. \(\dfrac{1}{x}-\dfrac{1}{x+1}\)

\(=\dfrac{1\left(x+1\right)}{x\left(x+1\right)}-\dfrac{1x}{\left(x+1\right)x}\)

\(=\dfrac{x+1}{x\left(x+1\right)}+\dfrac{-x}{x\left(x+1\right)}\)

\(=\dfrac{\left(x+1\right)-x}{x\left(x+1\right)}\)

6. \(\dfrac{1}{2x^2-10x}-\dfrac{1}{x-5}\)

\(=\dfrac{1}{2x\left(x-5\right)}-\dfrac{1}{x-5}\)

\(=\dfrac{1}{2x\left(x-5\right)}-\dfrac{1.2x}{2x\left(x-5\right)}\)

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

\(=\dfrac{1-2x}{2x\left(x-5\right)}\)

7. \(\dfrac{x-1}{x^2-1}.\dfrac{x+1}{x+3}\)

\(=\dfrac{\left(x-1\right)\left(x+1\right)}{\left(x^2-1\right)\left(x+3\right)}\)

\(=\dfrac{x^2-1}{\left(x^2-1\right)\left(x+3\right)}\)

8. \(\dfrac{2}{2x^2+10x}.\dfrac{x+5}{3x}\)

\(=\dfrac{2x\left(x+5\right)}{2x^2+10x.3x}\)

\(=\dfrac{2\left(x+5\right)}{2x\left(x+5\right)3x}\)

\(=\dfrac{2}{6x^2}=\dfrac{1}{3x^2}\)