What is the difference? \( \frac{x}{x^{2}-16}-\frac{3}{x-4} \) \[ \frac{x-3}{(x+5)(x-4)} \]

To tackle this question, let's first rewrite the expression clearly. The expression is: \[ \frac{x}{x^{2}-16} - \frac{3}{x-4} \] Now, notice that \( x^2 - 16 \) can be factored as \( (x-4)(x+4) \). This means that the first fraction can be rewritten: \[ \frac{x}{(x-4)(x+4)} - \frac{3}{x-4} \] To combine these fractions, we need a common denominator, which would be \( (x-4)(x+4) \). The second fraction becomes: \[ \frac{3}{x-4} \cdot \frac{x+4}{x+4} = \frac{3(x+4)}{(x-4)(x+4)} \] Now we can combine the two fractions: \[ \frac{x - 3(x + 4)}{(x-4)(x+4)} = \frac{x - 3x - 12}{(x-4)(x+4)} = \frac{-2x - 12}{(x-4)(x+4)} \] Finally, we can factor out \(-2\): \[ \frac{-2(x + 6)}{(x-4)(x+4)} \] This gives us the result. Now, just to address the second part, which is the expression: \[ \frac{x-3}{(x+5)(x-4)} \] So, when comparing the two results, you can conclude that \( \frac{-2(x + 6)}{(x-4)(x+4)} \) is not equivalent to \( \frac{x-3}{(x+5)(x-4)} \) as the numerators and denominators do not match up for simplification without specific values of \( x \). In summary, you have a rational expression comparison, yielding two distinct results. Fun tip: When dealing with rational expressions, always keep an eye out for factoring opportunities. It could save you a lot of time and lead you to an easier solution! Trust the factor! 😄