finding an equivilant expression.

Question

For x ≠ 0, x ≠ 1, and x ≠ −1 which of the expressions below is equivalent to

(x^−1) /(x^−2 − 1 )?

 

1) 1/(x^3 − x)

2) x/(1 − x^2) 

3) 1/(1 − x^2) 

4) 1/ (x − x^3) 

5) x/(1 − x)

 

I'm thinking number 5 but I'm not sure because I think it really should be x^2/x-1 but number five is the closest to that. please help me, I don't understand how to determine the answer.

Answer 1

Answer:  2) x/(1 − x^2)   because...

 

= [1/x] ÷ [1/x² - 1]  =   [1/x] ÷ [1/x² - x²/x²]  =  [1/x] ÷ [(1 - x²)/x²]  =

 

[1/x] • [x² / (1 - x²)]  =  x / (1-x²)

Answer 2

(x⁻¹) /(x⁻² − 1 )

 

you can rearrange the Eq. by eliminating the negative exponents.

 

x^-1 = 1/x

x^-2 = 1/x²

 

(x⁻¹) /(x⁻² − 1 ) = (1/x)/(1/x² - 1)

set x²/x² = 1

 

(1/x)/(1/x² - x²/x²) = (1/x)/[(1 - x²)/x²]

 

now flip and multiply

 

(1/x) * [x²/(1 - x²)] = x²/(x(1 - x²)

 

an x cancels out giving;

 

x/(1 - x²)

which is solution #2.