Show that each algebraic fraction simplifies to the given expression.

a) 2/x+1 + 5/x+2 = 3 simplifies to 3x² + 2x - 3 = 0

b) 3/4x+1 - 4/x+2 = 2 simplifies to 8x² + 31x + 2 = 0

c) 2/2x-1 - 6/x+1 = 11 simplifies to 22x² + 21x - 19 = 0

Hello,

a)

we suppose that x≠-1 and x≠-2

2 / (x+1) + 5 / (x+2) = 3

==>2 (x+2) + 5 (x+1) = 3 (x+1) (x+2)

==>7x+9=3x²+9x+6

==>3x²+2x+3=0

b)

we suppose that x≠-1/4 and x≠-2

3 / (4x+1) - 4 / (x+2) = 2

==>3 (x+2) - 4 (4x+1) = 2 (4x+1) (x+2)

==>-13x+2=8x²+18x+4

==>8x²+31x+2=0

c)

we suppose that x≠1/2 and x≠-1

2 / (2x-1) - 6 / (x+1) = 11

==>2 (x+1) - 6 (2x-1) = 11 (2x-1) (x+1)

==>-10x+8=22x²+11x-11

==>22x²+21x-19=0

a) 2/x+1 + 5/x+2 = 3 simplifies to 3x² + 2x - 3 = 0

2 (x+2) + 5 (x+1) = 3 (x+2) (x+1)

2x + 4 + 5x + 5 = (3x+6) (x+1)

7x + 9 = 3x² + 3x + 6x + 6

7x + 9 = 3x² + 9x + 6

0 = 3x² + 9x - 7x + 6 - 9

0 = 3x² + 2x - 3

b) 3/4x+1 - 4/x+2 = 2 simplifies to 8x² + 31x + 2 = 0

3 (x+2) - 4 (4x+1) = 2 (x+2) (4x+1)

3x + 6 - 16x - 4 = (2x+4) (4x+1)

-13x + 2 = 8x² + 2x + 16x + 4

0 = 8x² + 18x + 13x + 4 - 2

0 = 8x² + 31x + 2

c) 2/2x-1 - 6/x+1 = 11 simplifies to 22x² + 21x - 19 = 0

2 (x+1) - 6 (2x-1) = 11 (x+1) (2x-1)

2x + 2 - 12x + 6 = (11x+11) (2x-1)

-10x + 8 = 22x² - 11x + 22x - 11

0 = 22x² + 11x + 10x - 11 - 8

0 = 22x² + 21x - 19