State the restrictions on the variable and classify the restrictions as removable or nonremovable.

Question

(x−3)/(x+4)÷(x^2−1)/x
 

Question 1
What is being asked in the problem and what does that mean? What plan am I going to try? (1 sentence)

Question 2
I will write out my response to the question, explaining my answer and what it means. (1 sentence)

(solve step by step at the end)

Answer 1

The problem is asking for the restrictions on the variable in the given expression and to classify them as removable or nonremovable.
I will analyze the expression to identify any values of x that would make it undefined, which would be the restrictions on the variable, and then determine if they are removable or nonremovable.

Step by step solution:

1. The expression is: (x-3)/(x+4) ÷ (x^2-1)/x.
2. Simplify the expression inside the division sign: (x-3)/(x+4) ÷ (x^2-1)/x = (x-3)/(x+4) ÷ [(x+1)(x-1)]/x.
3. Rewrite the expression as multiplication by flipping the second fraction: (x-3)/(x+4) * (x/x) / (x+1)(x-1).
4. Simplify: (x-3)/(x+4) * 1 / (x+1)(x-1).
5. Determine any values of x that would make the expression undefined:
- x+4 ≠ 0, so x ≠ -4
- x ≠ 0
- x+1 ≠ 0, so x ≠ -1
- x-1 ≠ 0, so x ≠ 1
6. The restrictions on the variable are x ≠ -4, x ≠ 0, x ≠ -1, x ≠ 1. These restrictions are nonremovable as they are necessary for the expression to be defined.

Answer 2

you cannot eliminate the x factor. Where do you justify multiplying by x/x?
The final expression is
(x-3)/(x+4) * x / (x+1)(x-1) = x(x-3) / (x+4)(x^2-1)

everything else you said is ok.
x=0 still must be eliminated, since the original equation requires it.

Answer 3

You are correct, I made a mistake in multiplying by x/x. Thank you for pointing that out.
The correct final expression is (x-3)/(x+4) * x / (x+1)(x-1) = x(x-3) / (x+4)(x^2-1).
And yes, x=0 should still be eliminated as it is required to make the original equation defined. Thank you for the clarification.