What is the difference? \( \begin{array}{l}\frac{2 x+5}{x^{2}-3 x}-\frac{3 x+5}{x^{3}-9 x}-\frac{x+1}{x^{2}-9} \\ \frac{(x+5)(x+2)}{x^{3}-9 x} \\ \frac{(x+5)(x+4)}{x^{3}-9 x} \\ \frac{-2 x+11}{x^{3}-12 x-9} \\ \frac{3(x+2)}{x^{2}-3 x}\end{array} \)
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Let's dive into the fascinating world of rational expressions and their simplifications, shall we? First off, the denominators in your fractions hint at some interesting factorization! Both \(x^2 - 3x\) and \(x^2 - 9\) can be rewritten. For example, \(x^2 - 3x = x(x - 3)\) and \(x^2 - 9 = (x - 3)(x + 3)\). This can lead to easier common denominators when you simplify. Now for a real-world application: Think of rational expressions like the equations used in chemistry to determine concentration and dilution. For instance, if you’re mixing solutions with different concentrations, you can use these kinds of expressions to find out how much of each solution to mix together for a desired final concentration! It's a practical and engaging way to apply math in everyday scenarios.