Question

How do I find the restrictions on x if there are any?
`(1)/(x - 1) = (5)/(x - 10)`

Answer 1

We have the expression:

`(1)/(x - 1)=(5)/(x - 10)`

When we have rational functions, where the denominator is a function of x, we have a restriction for the domain for any value of x that makes the denominator equal to 0.

That is because if the denominator is 0, then we have a function f(x) that is a division by zero and is undefined.

If we have a value that makes f(x) to be undefined, then this value of x does not belong to the domain of f(x).

Expression:

`\begin{gathered} (1)/(x-1)=(5)/(x-10) \\ (x-1)/(1)=(x-10)/(5) \\ x-1=(x)/(5)-(10)/(5) \\ x-1=(1)/(5)x-2 \\ x-(1)/(5)x=-2+1 \\ (4)/(5)x=-1 \\ x=-1\cdot(5)/(4) \\ x=-(5)/(4) \end{gathered}`

Answer: There is no restriction for x in the expression.