Equivalent equations

[mathwannabe]

Hello everybody

I got this problem that has gotten me confused:

1) Are the following two equations equivalent:

\(\displaystyle x^2(1-\dfrac{1}{\sqrt{-x}})=4-\dfrac{4}{\sqrt{-x}}\) AND \(\displaystyle x^2=4\)

I have found that the solutions for the first equation \(\displaystyle -2\) and \(\displaystyle 2\) are also the solutions for the second equation, but, for \(\displaystyle x=2\) the expression \(\displaystyle 1-\dfrac{1}{\sqrt{-x}}\) is not defined in \(\displaystyle R\). The text of the problem does not explicitly state restrictions to any particular set of numbers. What should my answer be? Are they equivalent or not?

I was guided by the theorem that \(\displaystyle f(x)*r(x)=g(x)*r(x)\) if, and only if \(\displaystyle r(x)=0\) or \(\displaystyle f(x)=g(x)\)

 

[mathwannabe]

Unless I'm missing something, this is VERY simple:
Let k=SQRT(-x)

x^2(k - 1) / k = (4k - 4) / k

x^2(k - 1) = 4(k - 1)

x^2 = 4

Yes, but then for \(\displaystyle x=2\) we will have a negative under the square root, which is not defined in the set of real numbers. That's what's confusing me...

 

[mathwannabe]

Well, if there are no restrictions on the values of the equations (or on f, g, or r), then I would say they are equivalent. IIRC, equations are equivalent if they have the same solutions, which these do. But if this is in preparation for an exam or a class, I would ask for clarification from the examiner/instructor. Because if the equations are real valued, then you are correct, they are not equivalent.

Yes, thanks for the post.

 

[lookagain]

Hello everybody

I got this problem that has gotten me confused:

1) Are the following two equations equivalent:

\(\displaystyle x^2(1-\dfrac{1}{\sqrt{-x}})=4-\dfrac{4}{\sqrt{-x}}\) AND \(\displaystyle x^2=4\)

I have found that the solutions for the first equation \(\displaystyle -2\) and \(\displaystyle 2\)

are also the solutions for the second equation, but, for \(\displaystyle x=2\) the expression

\(\displaystyle 1-\dfrac{1}{\sqrt{-x}}\) is not defined in \(\displaystyle R\).

The text of the problem does not explicitly state restrictions to any particular set of numbers.
What should my answer be? Are they equivalent or not?

They are not equivalent equations, in part, because x = -1 is a solution to the
first equation (check this), whereas x = -1 is not a solution to \(\displaystyle x^2 = 4.\)

 

[mathwannabe]

They are not equivalent equations, in part, because x = -1 is a solution to the
first equation (check this), whereas x = -1 is not a solution to \(\displaystyle x^2 = 4.\)

Yes, thank you... I have been very sloppy on this one...

 

[lookagain]

1) Are the following two equations equivalent:

\(\displaystyle x^2(1-\dfrac{1}{\sqrt{-x}})=4-\dfrac{4}{\sqrt{-x}} \ \ AND \ \ x^2=4\)

Suppose there was a related question asked:

"What are the x-values of the points of intersection of \(\displaystyle y \ = \ x^2\bigg(1 - \dfrac{1}{\sqrt{-x}}\bigg) \ \ with \ \ y \ = \ 4 - \dfrac{4}{\sqrt{-x}}?"\)


The answers would be x = -2 and x = -1. The real domains for both are \(\displaystyle (-\infty, 0),\)
because the radicand, -x, must be less than zero.


Then x = 2 would not be an x-value, because there are no points of intersection to the right
of the y-axis, for instance. The graphs don't meet over there.