Question

How do I find the restrictions on x if there are any? D%20%20%5Cfrac%7B5%7D%7Bx%20-%2010%7D%20" id="TexFormula1" title=" \frac{1}{x - 1} = \frac{5}{x - 10} " alt=" \frac{1}{x - 1} = \frac{5}{x - 10} " align="absmiddle" class="latex-formula">

Answer 1

We have the expression:

$\frac{1}{x - 1}=\frac{5}{x - 10}$

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:

$\begin{gathered} \frac{1}{x-1}=\frac{5}{x-10} \\ \frac{x-1}{1}=\frac{x-10}{5} \\ x-1=\frac{x}{5}-\frac{10}{5} \\ x-1=\frac{1}{5}x-2 \\ x-\frac{1}{5}x=-2+1 \\ \frac{4}{5}x=-1 \\ x=-1\cdot\frac{5}{4} \\ x=-\frac{5}{4} \end{gathered}$

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