My last maths question today is this....
"Explain what is wrong with this argument"
is equivalent to (divided both sides by (x+1)
So the solution to is
Which equation? There is nothing undefined about x(x + 1) = 0. On the other hand, dividing both sides by x + 1 is in fact a statement that says,Ok thanks , so for both solutions the equation is undefined because if I use the value of x=-1 I will be dividing by zero which is forbidden?(same with x=0 forbidden)
For all x, x(x + 1) = 0 iff x(x + 1)/(x + 1) = 0
This statement is false for x = -1.
That is why, when simplifying an equation, it is important to write not just a sequence of equations E1 = E'1, E2 = E'2, ..., but also the logical relationship between the them, e.g., E1 = E'1 <=> E2 = E'2 => E3 = E'3, ... In the example above,
x(x + 1) = 0 <= x(x + 1)/(x + 1) = 0
is OK, but
x(x + 1) = 0 <=> x(x + 1)/(x + 1) = 0