Question

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

(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)

Answers

GPT 3.5
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.
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.
GPT 3.5
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.

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