Simplify x/6x-x^2

I know that the answer is 1/6-x but the answers look like this:
A. 1/6-x where x is not equal to 0,6
B. 1/6-x where x is not equal to 6

Please help me with understand why is says x is not equal to, I have like 5 problems just like this so I really need to understand this, thanks so much.

the denominator is 6x-x^2 = x(6-x)

division by zero is not defined, and since the denominator is zero at x=0,6
those two values must be excluded from consideration. So, A

Thank you, so is simplify 5x^3/7x^3+x^4 = 5/7+x where x is not equal to 0,-7?

yes, as I noted in your earlier post.

To understand why the expression x/6x - x^2 is written as 1/6 - x with the condition x is not equal to 0 or 6, let's break down the steps to simplify the expression.

First, we notice that the denominator of the first term, 6x, can be factored as 6 * x. So the expression becomes:

x / (6 * x) - x^2

Now, we can simplify the first term by canceling out the common factor of x:

1/6 - x^2

Next, we can factor out a negative sign from the x^2 term:

1/6 - (1 * x^2)

1/6 - (x^2)

Since we can't factor the numerator any further, it remains as 1.

Now we have the simplified expression: 1/6 - x^2.

The next step is to check for any restrictions or values that make the expression undefined. In this case, we have a fraction in the form of 1/6 - x^2.

Notice that the denominator of the fraction is a constant (6), so it is always defined as long as x is a real number. However, the x^2 term in the expression might introduce restrictions.

To identify any restrictions, we need to consider the original expression before simplifying: x/6x - x^2.

If we recall the definition of division, dividing any number by zero is undefined. Hence, the expression x/6x will be undefined when 6x is equal to zero (i.e., x = 0). Consequently, this restriction also applies to the simplified expression 1/6 - x^2.

Furthermore, since we initially factorized the denominator of the first term as 6 * x, we should consider restrictions from both factors. Therefore, to account for x = 0, we must include it as a restriction.

In the given answer choices, A and B both state that x should not be equal to 0. This is correct since x = 0 would make the expression undefined.

Additionally, answer choice B states that x should not be equal to 6. However, this restriction does not arise from simplifying the expression itself. It might be an additional condition given in the context of the problem or an extraneous solution that should be excluded from the solution set.

In conclusion, the simplified expression is 1/6 - x^2, and x should not be equal to 0 for it to be defined.