1. Исследуйте функцию y = x^2-3 / x-2

(а+3)/(а-1)  +  (а+5)/(2-2а) = 
= (а+3)/(а-1) + (а+5)/ -2(а-1) =
= (а+3)/(а-1)  -   (а+5)/2(а-1)=
=  (2(а+3)  - (а+5))/ 2(а-1)=
=  (2а+6 -а -5) /2(а-1)=
= (а+1)/2(а-1)  =
= (а+1)/(2а-2)

4m/(m²-n²)   - 4/(m+n) =
= (4m - 4(m-n) ) /  (m+n)(m-n)   =
= (4m - 4m +4n) / (m+n)(m-n) =
= 4n/(m²-n²)

(2x-y)/(2x+y)  +    y/(4x+2y) =
= (2x-y)/(2x+y) +   y/2(2x+y)=
= ( 2(2x-y) +y)  /  2(2x+y)=
= (4x-2y+y) / 2(2x+y)=
=  (4x-y)/ 2(2x+y)  =
= (4x-y) / (4x+2y)

(x²+y²)/(x²-y²)   -  (x-y)/(x+y) =
=(x²+y² - (x-y)(x-y) ) /    (x-y)(x+y) =
= (x²+y² - (x² -2xy +y²)) /  (x-y)(x+y) =
= (x² +y²-x²+2xy -y²) / (x-y)(x+y) =
= 2xy / (x²-y²)

4/(a²-9)    -2/(a²+3a) =
= 4/(a-3)(a+3)       -    2/ a(a+3)  =
=  (4a -2(a-3))  /   (a(a-3)(a+3) )=
=  (4a -2a+6)   /   a(a-3)(a+3) =
=   (2a+6)   /  a(a-3)(a+3)  =
=   2(a+3) / a(a-3)(a+3) =
= 2/ a(a-3)  

1/(2m -4)   +  m/(4-m²) = 
= 1/ 2(m-2)       -     m/(m-2)(m+2)  = 
= (m+2  - 2m)  /  2(m-2)(m+2)  =
=   (-m+2) /  2(m-2)(m+2) =
=    -   (m-2)/  2(m-2)(m+2) =
=  -   1/ 2(m+2) =  
= -   1/(2m+4)

(4x²+4)/(16x²-9)      - (x²+1)/(4x² -3x) =
= (4x²+4) /  (4x-3)(4x+3)       -     (x²+1) / x(4x-3)  =
=  (x(4x²+4)  -  (x²+1)(4x -3))  /  ( x(4x-3)(4x+3) )=
=  (4x³ +4x  - (4x³ -3x² +4x-3))  / (x(4x-3)(4x+3)) =
= ( -3x²  +3 )  /   (x(4x-3)(4x+3)) =
=  -     (3x² -3)/  (x (16x² -9 )) = 
=  -     (3x² - 3) / (16x³ -9x)

b/(3b-1)    +      (3b² -1)/ (1-9b²)  = 
=  b/(3b-1)    -    (3b² -1) /  (3b-1)(3b+1)  = 
=  (b(3b+1)   - (3b² - 1) ) /  (3b-1)(3b+1) =
=  (3b² + b - 3b² + 1) /   (3b-1)(3b+1) =
=  (b+1)  /(9b² - 1)

1)y/y^2-9 - 3/9-y^2 = y /y^2-9 - (-3/y^2-9) = y /y^2-9 + 3/y^2-9 = y+3/(y-3)(y+3)= 1/y-3

2)2x/x^2-4 - 4/4 - x^2 =2x/x^2-4 - (-4/x^2-4) = 2x+4/x^2-4 = 2(x+2)/(x-2)(x+2)= 2/x-2

3)x2/x-7 + 7x/7-x = x^2/x-7 - 7x/x-7 = x^2-7x/x-7 = x(x-7)/x-7= x

4)7/x^2-9x + 3x-7/x^2-9x =7+3x-7/x^2-9x= 3x/x^2-9x

5)5y-19/11y + 6y+8/11y =5y-19+6y+8/11y= 11y - 27/11y

6)y+3/y-3 - 9-y/y-3 =y+3-9-y/y-3=-3/y-3

7)12a+b/28a - b-30a/28a =12a+b-b-30a/28a = -28a/28a = -1

8)6a-4/2 - 2-3a/2 =6a-4-2-3a/2 = 3a+6/2 = 3(a+2)/2

9)b/a-b - a/a-b =-a/a-b - b/a-b = -(a-b)/a-b = -1

10)5a/a-b - 5b/a-b =5a-5b/a-b = 5(a-b)/a-b = 5

11)m^2/m-5 - 25/m-5 =m^2-25/m-5 = (m-5)(m+5)/m-5= m+5

12)a^2-47/a+7 - 2/a+7 =a^2-47-2/a+7 = a^2-49/a+7 = (a-7)(a+7)/a+7=a-7

13)m/m^2-n^2 - n/m^2-n^2 =m-n/m^2-n^2= m-n/(m-n)(m+n)=1/m+n

14)m2/m+n - n^2/m+n =m^2-n^2/m+n = (m-n)(m+n)/m+n = m-n

15)n^2/3n+3m - m^2/3n+3m =n^2-m^2/3n+3m = (n-m)(n+m)/3(n+m) = n-m/3