Multiplying by a variable in an equation: (3x+7)/x=10

[otoole.timothy]

(3x+7)/x=10 is Q17 from Drill 3 of MG FOM 6th Edition Equations chapter. It's on page 277.

I understand the solution to this equation but when I initially worked on it I was hesitant to multiply both sides by x to get rid of the variable in the denominator because I did not know for sure that x was not zero or whether the variable represented a positive or negative number.

My two questions are:

1) Is it safe to assume the x is not zero because if it were the equation would be undefined and it really would not be something that makes sense in the context of a GMAT problem?

2) I think I may have confused the principles of multiplying and dividing variables in an equation vs. in an inequality. In an equation can I freely multiply and divide by variables when I do not know the sign? I know in inequalities I need to be careful because multiplying or dividing by a negative number changes the direction of the sign.

Thanks in advance for your assistance.

[Sage Pearce-Higgins]

In answer to your questions:

1) This equation would make no sense if x = 0, so you're safe to assume that. However, remember that this is a question from the FoM, so it's not a real GMAT problem. Problems in the GMAT exam always take care to state that 'x does not = 0' or similar when required.
2) You're right. It's only with inequalities that we have to be careful about multiplying by negative numbers (or variables that could be either positive or negative). For equations we're free to jump from x = 5 to -x = -5.

To get a deeper understanding of equations, try plugging the answer back into an equation after you've solved it to check that the answer works.