- Factor each numerator and denominator: \(\frac{x+2}{(x-4)(x+4)} \div \frac{2(x+2)}{(x+4)(x-1)}\).
- Rewrite the division as multiplication by the reciprocal: \(\frac{x+2}{(x-4)(x+4)} \cdot \frac{(x+4)(x-1)}{2(x+2)}\).
- Cancel out the common factors \((x+2)\) and \((x+4)\).
- Obtain the simplified result: \(\boxed{\frac{x-1}{2(x-4)}}\).
Explanation
- Factoring the rational expressions
To simplify the expression \(\frac{x+2}{x^2-16} \div \frac{2x+4}{x^2+3x-4}\), we first factor each numerator and denominator completely. The denominator of the first fraction is a difference of squares: \(x^2 - 16 = (x-4)(x+4)\). The numerator of the second fraction has a common factor: \(2x+4 = 2(x+2)\). The denominator of the second fraction is a quadratic trinomial: \(x^2 + 3x - 4\). We look for two numbers that multiply to \(-4\) and add to \(3\), which are \(4\) and \(-1\). Thus, \(x^2 + 3x - 4 = (x+4)(x-1)\).
- Rewriting the division as multiplication
Substituting these factored forms into our original division problem, we have:
\[ \frac{x+2}{(x-4)(x+4)} \div \frac{2(x+2)}{(x+4)(x-1)} \]
Recalling the rule for dividing fractions, which is to multiply by the reciprocal of the divisor, we get:
\[ \frac{x+2}{(x-4)(x+4)} \cdot \frac{(x+4)(x-1)}{2(x+2)} \]
- Simplifying the expression
Now, we combine the expressions into a single fraction:
\[ \frac{(x+2)(x+4)(x-1)}{(x-4)(x+4) \cdot 2(x+2)} \]
Assuming \(x
eq 4, -4, 1, -2\), we can cancel the common factors \((x+2)\) and \((x+4)\) from both the numerator and the denominator. This leaves us with:
\[ \frac{x-1}{2(x-4)} \]
- Conclusion
Comparing our result \(\frac{x-1}{2(x-4)}\) with the given choices, we identify that it matches the third option: \(\frac{x-1}{2(x-4)}\). The final answer is \(\boxed{\frac{x-1}{2(x-4)}}\).
Examples
Rational expressions are frequently used in engineering and physics, for example, when calculating the combined resistance of resistors in a parallel circuit or determining the average velocity over varying time intervals. Simplifying these expressions algebraically, as done here, is a fundamental skill that allows for faster and more efficient modeling of real-world systems, turning complex ratios into manageable formulas.