Determine the Validity: Is x^2 - 81 Over (x-9)(x+9) Equal to 1?

Does the following equation have a true or false value?

$\frac{x^2-81}{(x-9)(x+9)}=1$

To solve this problem, we'll follow these steps:

• Step 1: Recognize and factor the expression in the numerator.
• Step 2: Simplify the fraction by canceling common factors.
• Step 3: Determine restrictions on the variable $x$.
• Step 4: Analyze if and when the equation holds true.

Now, let's work through each step:

Step 1: The numerator $x^2 - 81$ can be factored as a difference of squares: $(x-9)(x+9)$.

Step 2: Substitute this factorization into the equation:
$\frac{(x-9)(x+9)}{(x-9)(x+9)} = 1$.

Step 3: Simplify the fraction by canceling the common terms, giving $1 = 1$, which is always true, except where the expression is undefined.

Step 4: The expression is undefined when the denominator is zero, i.e., when $x - 9 = 0$ or $x + 9 = 0$. Thus, $x \neq 9$ and $x \neq -9$.

In conclusion, the given equation $\frac{x^2-81}{(x-9)(x+9)}=1$ is True only when $x \ne \pm9$.