Brilliant's advertising problem

I joined brilliant.org after I saw an advertising which asks for the solution of $\frac{x}{x-1}=\frac{1}{x-1}.$ Now I'm curious what the official answer to this question is?

Easy Math Editor

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Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

The equation has no solutions.............

Aaghaz Mahajan - 3 years ago

One thing that may help is looking at the graph of each side:

Graph at Wolfram Alpha

Notice they seem to be converging to a vertical asymptote at $x =1 ,$ although it's unclear if one graph might "catch up" to the other. Whatever is happening, if there's an intersection it seems to be at "a little more than 1" or "a little less than 1". Let's use $q$ for the "little" part and consider the solution $1 + q .$

This makes the two sides $\frac{1+q}{1+q-1}$ and $\frac{1}{1+q-1}$ which simplify to $\frac{1+q}{q}$ and $\frac{1}{q} .$ Now it's a little easier to see that the left side will never quite match the right side - it's always off by that slight factor $+q$ in the numerator.

Jason Dyer Staff - 2 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$