James R. answered • 05/29/19

Take Advantage of 30+ years of Technology, Science and Math

The challenge here is to determine how best to apply "precedence rules" or "order of operations." You can Google these terms for lots of tutorials on the subject. The problem you presented has perhaps been designed to be tricky in order to promote thinking. The correct order if operations is not obvious. Also, I assume you are making no distinction between lower case and upper case variables, as happens in some programming languages, hence I hereby assert: X = x.

Given my assumption. this is the most likely way I would interpret this problem, reading from left to right:

(a) (x/(x-2)) + ((x-1)/(x+1)) = -1

In this interpretation, I would think that the author had simply been sloppy and not paying attention the the confusion being caused by not being careful when grouping expressions. Notice how useful parentheses are in helping to understand the true meaning of the expressions.

The most __unlikely__ way to interpret this problem would be:

(b) (x/x) - 2 + x - (1/x) + 1 = -1

I consider this interpretation as unlikely based on my experience in reading and writing mathematical formulas, especially as they relate to physics and electronics. Other than being the result of prior simplification steps, I find it hard to imagine a useful equation being written in this manner. Hence. I consider (a) to be the best interpretation, and (b) to be the worst. There are others in-between, such as (x/(x-2)) + x - (1/x) + 1 = -1.

Note that both (a) and (b) follow correct order of operations, and are valid interpretations of the original problem. I reject, however, (b) as a real-world sensible example, and therefore choose (a).

If you accept my original assertion, and agree with my logic so far, then the correct solution to the equation given in the problem, is the solution to (a):

Here is a handy website: https://www.tiger-algebra.com/

Plugging in the equation: https://www.tiger-algebra.com/drill/(x/(x-2))+(x-1)/(x+1))=-1/

The two solutions found are x = 1 and x = 0. (Be careful when simplifying by hand: I originally did so, and also found a false solution of x = -1. Do you know why this cannot be a solution?)

Congratulations on presenting a very interesting and challenging problem!