Question

Solve 4 over x minus 4 equals the quantity of x over x minus 4, minus four thirds for x and determine if the solution is extraneous or not. (1 point)

Answer 1

I'm thinking this is what the problem looks like: $\frac{4}{x-4}= \frac{x}{x-4}- \frac{4}{3}$.  The first thing to do is to move the $\frac{x}{x-4}$ over to the other side because it has a common denominator with the other side.  Doing that and at the same time combining them over their common denominator looks like this: $\frac{4-x}{x-4}= -\frac{4}{3}$.  The best way to solve for x now is to cross-multiply to get 3(4-x)=-4(x-4).  Distributing through the parenthesis is 12 - 3x = -4x + 16.  Solving for x gives us x = 4.  Of course when we sub a 4 back in for x we get real problems, don't we?  Dividing by zero breaks every rule in math that there ever was! So, yes, the solution is extraneous.