how to this problem

Question

x     -     2        =       8

x-1        1-x²                   x+1

Answer 1

Need a least common denominator its

(x-1)(x+1)=x²-1

 

multiply through by the LCD

x•(x+1)+2=8•(x-1)

 

Note the sign change on two since its denominator was -LCD

 

 

x² +x +2=8x-8

x²-7x+10=0

 

factor

(x-5)(x-2)=0

 

x=5, 2

 

 

Answer 2

x / (x - 1) - 2 /( 1 - x²) = 8 / (x + 1)               Multiply second fraction numerator and denominator by -1

x / (x - 1) + 2 / (x² - 1) = 8 / (x + 1)              x² - 1 = (x + 1)(x - 1). Multiply both sides

x(x + 1) + 2 = 8(x - 1)

x² + x + 2 = 8x - 8

x² -7x + 10 = 0

(x - 2)(x - 5) = 0

 

{2, 5}

 

 

Answer 3

x/_x-1 - ²/_1-x² = ⁸/_x+1

 

The first thing we need to do is find a common denominator. We need to factor out that middle term to see what parts make it up:

-x² + 1 = (-x + 1)(x + 1) → -(x - 1)(x + 1)

Since we are subtracting a negative, we can flip that middle sign to a positive as well:

 

^x/_x-1 + ²/_{(x - 1)(x + 1)} = ⁸/_x+1

Now back to the LCD. The first fraction needs an x+1 in it's denominator, and the answer an x-1, so we multiply by essentially 1/1 in each of these cases:

^x(x+1)/_{(x - 1)(x + 1)} + ²/_{(x - 1)(x + 1)} = ^8(x-1)/_{(x - 1)(x + 1)}

 

Now while the denominator's important, I'm just going to focus on the numerator right now to save eye strain:

x(x + 1) + 2 = 8(x - 1)

x² + x + 2 = 8x - 8

Move everything to one side to get our quadratic equation:

x² - 7x + 10 = 0

Factor. Since the middle term is negative and last positive, we know both factors will have a - sign:

(x - 2)(x - 5) = 0

x = 2, 5