1. Find `(1)/(x+1) + (x)/(x-1)`
Solution:
`=1/(x+1)+x/(x-1)`
`=(1 xx (x-1)+x xx (x+1))/((x+1)(x-1))`
`=((x-1)+(x^2+x))/((x+1)(x-1))`
`=(2x-1+x^2)/((x+1)(x-1))`
2. Find `(1)/(x+1) - (x)/(x-1)`
Solution:
`=1/(x+1)-x/(x-1)`
`=(1 xx (x-1)-x xx (x+1))/((x+1)(x-1))`
`=((x-1)-(x^2+x))/((x+1)(x-1))`
`=(-1-x^2)/((x+1)(x-1))`
3. Find `(x-1)/(x+1) * (x)/(x-1)`
Solution:
`=(x-1)/(x+1)×x/(x-1)`
`=x/(x+1)`
4. Find `(1)/(x-1) / (x)/(x-1)`
Solution:
`=1/(x-1)/(x/(x-1))`
`=1/(x-1)×(x-1)/x`
`=1/x`
5. Find `(1)/(x-1) + (x)/(x-1) - (1)/(x+1)`
Solution:
`=1/(x-1)+x/(x-1)-1/(x+1)`
`=(1 xx (x+1)+x xx (x+1)-1 xx (x-1))/((x-1)(x+1))`
`=((x+1)+(x^2+x)-(x-1))/((x-1)(x+1))`
`=(x+2+x^2)/((x-1)(x+1))`
6. Find `(1)/(x-1) / (x)/(x-1) * (1)/(x+1)`
Solution:
`=1/(x-1)/(x/(x-1))×1/(x+1)`
`=1/(x-1)×(x-1)/x×1/(x+1)`
`=1/(x^2+x)`
7. Find `(x-1)/(x+1) - (x+1)/(x-1) + (4x)/(x^2-1)`
Solution:
`=(x-1)/(x+1)-(x+1)/(x-1)+(4x)/(x^2-1)`
`=(x-1)/(x+1)-(x+1)/(x-1)+(4x)/((x-1)(x+1))`
`=((x-1) xx (x-1)-(x+1) xx (x+1)+4x)/((x+1)(x-1))`
`=((x^2-2x+1)-(x^2+2x+1)+4x)/((x+1)(x-1))`
`=0/((x+1)(x-1))`
`=0`
8. Find `(x+2)/(x^2-4x+3) - (x+3)/(x^2-3x+2) + (x+1)/(x^2-5x+6)`
Solution:
`=(x+2)/(x^2-4x+3)-(x+3)/(x^2-3x+2)+(x+1)/(x^2-5x+6)`
`=(x+2)/((x-1)(x-3))-(x+3)/((x-1)(x-2))+(x+1)/((x-2)(x-3))`
`=((x+2) xx (x-2)-(x+3) xx (x-3)+(x+1) xx (x-1))/((x-3)(x-2)(x-1))`
`=((x^2-4)-(x^2-9)+(x^2-1))/((x-3)(x-2)(x-1))`
`=(4+x^2)/((x-3)(x-2)(x-1))`
9. Find `(x^2+5x+4)/(x^2-4) * (x^2-5x+6)/(x^2-16) / (x^2-2x-3)/(x^2-2x-8)`
Solution:
`=(x^2+5x+4)/(x^2-4)×(x^2-5x+6)/(x^2-16)/((x^2-2x-3)/(x^2-2x-8))`
`=((x+1)(x+4))/((x-2)(x+2))×((x-2)(x-3))/((x-4)(x+4))/(((x+1)(x-3))/((x+2)(x-4)))`
`=((x+1)(x+4))/((x-2)(x+2))×((x-2)(x-3))/((x-4)(x+4))×((x+2)(x-4))/((x+1)(x-3))`
`=1`
This material is intended as a summary. Use your textbook for detail explanation.
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