Here are my answers. Can you check if I got the right answers? Thank you!
1/(x+h) - 1/x= -h/(x^2+xh)
(2/x^2)/(10/x^5)= (x^3)/5
[1/(3+x) - 1/3]/x= -1/(9+3x)
2x/(x^2-6x+9) - 1/(x+1)- 8/(x^2-2x-3)= (x^2+15)/(x^3-5x^2+3x+9)
first 3 look ok
In the last one I would not have expanded the denominator, leave it as
(x+1)(x-3)^2
To verify if your answers are correct, I will go through each question one by one:
1. For the equation 1/(x+h) - 1/x = -h/(x^2 + xh)
To check your answer, you can start by cross-multiplying:
x - (x + h) = -h(x)
Simplifying:
x - x - h = -h(x)
-h = -h(x)
Dividing both sides by -h:
1 = x
Therefore, the solution is x = 1.
2. For the equation (2/x^2)/(10/x^5) = (x^3)/5
First, simplify the left side of the equation:
(2/x^2)/(10/x^5) = (2/x^2) * (x^5/10)
Next, cancel out the x^2 term:
2 * x^5 / (10 * x^2)
Simplifying further:
x^3 / 5
Therefore, your answer is correct as (x^3)/5.
3. For the equation [1/(3+x) - 1/3] / x = -1/(9+3x)
Begin by simplifying the left side of the equation:
[1/(3+x) - 1/3] / x = [3 - (3+x)] / (3+x) * x
Simplify further:
-x / (3+x) * x = -x^2 / (3+x)
Therefore, your answer is correct as -1 / (9+3x).
4. For the equation 2x / (x^2 - 6x + 9) - 1 / (x + 1) - 8 / (x^2 - 2x - 3) = (x^2 + 15) / (x^3 - 5x^2 + 3x + 9)
Your answer seems to have expanded the denominator, but it is not necessary.
The original equation can be written as:
2x / (x^2 - 6x + 9) - 1 / (x + 1) - 8 / [(x - 3)(x + 1)] = (x^2 + 15) / (x^3 - 5x^2 + 3x + 9)
From your answer, expand the denominator and cross-multiply to check if it simplifies to this equation.