Solve these equations. Please Please Help?

1.  

7/(x + 2) + 5/(x - 2) = (10x - 2)/(x^2 - 4)

7(x - 2) + 5(x + 2) = 10x - 2

2x = 2

x = 1

2.

x/(x - 2) + 1/(x - 4) = 2/(x^2 - 6x + 8)

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

x^2 - 3x - 4 = 0

(x - 4)(x + 1) = 0

x = -1

x = 4

3.

(2x - 4)/(x^2 - 10x + 16) = 2/(x + 2)

2(x^2 - 10x + 16) = (x + 2)(2x - 4)

2x^2 - 20x + 32 = 2x^2 - 8

20x - 40 = 0

x = 2

4.

2/(x^2 - x) = 1/(x - 1)

x^2 - x = 2x - 2

x^2 - 3x + 2 = 0

x^2 - 2x - x + 2 = 0

x(x - 2) - (x - 2)

(x - 2)(x - 1) = 0

x = 1

x = 2

5.

2x/(x - 3) = (3x/(x^2 - 9)) + 2

2x(x + 3) = 3x + 2(x^2 - 9)

6x = 3x - 9

x = -3

(4 (3 x - 1))/((x - 2) (x + 2)) = (2 (5 x - 1))/((x - 2) (x + 2))

12x-4=10x-2

2x=2

x=1

All these questions are similar. Adjust every term to have the same denominator, noting the difference of two squares. Here is the first one.

7/(x + 2) + 5/(x – 2) = (10x – 2)/(x^2 – 4)

[7(x – 2) + 5(x – 2) – 10x + 2]/(x^2 – 4) = 0

[7x – 14 + 5x – 10 – 10x + 2] = 0

2x – 22 = 0

x = 11

Copy that idea when you do the rest.

I'll do the first one.  They are all more or less the same with different values and expressions.

You have:

7 / (x + 2) + 5 / (x - 2) = (10x - 2) / (x² - 4)

We want to multiply both sides by the LCD of the three fractions in order to get rid of all fractions.  To determine what the LCD is we need to factor anything in the denominator.  So that gives us:

7 / (x + 2) + 5 / (x - 2) = (10x - 2) / [(x + 2)(x - 2)]

We can see that the LCD is now (x + 2)(x - 2). 

Note now that x cannot be -2 or 2 otherwise, we end up dividing by zero.  If we get that as an answer we have to throw it out which could mean there are no real solutions.

If we multiply all terms by that expression and cancel out the common factors we end up with:

7(x - 2) + 5(x + 2) = 10x - 2

Now we can expand and simplify the left side:

7x - 14 + 5x + 10 = 10x - 2

12x - 4 = 10x - 2

Now move terms with "x" to one side and all constants to the other:

2x = 2

And divide:

x = 1

This doesn't match our list of answers we have to throw out (-2 and 2) so we keep it.

To check, substitute this value in the original equation and simplify.  Both sides should be the same:

7 / (x + 2) + 5 / (x - 2) = (10x - 2) / (x² - 4)

7 / (1 + 2) + 5 / (1 - 2) = (10 * 1 - 2) / (1² - 4)

7 / 3 + 5 / (-1) = (10 - 2) / (1 - 4)

7 / 3 - 5 = 8 / (-3)

7 / 3 - 5 = -8 / 3

Multiply both sides by 3:

7 - 15 = -8

-8 = -8

TRUE

So our answer of x = 1 is the correct answer.