- Combine the fractions since they share a common denominator: \(\frac{-x-2-5x+4}{9x^2-1} = \frac{-6x+2}{9x^2-1}\).
- Factor the numerator as \(-2(3x-1)\) and the denominator as \((3x-1)(3x+1)\).
- Cancel the common factor \((3x-1)\) to simplify the expression to \(-\frac{2}{3x+1}\).
- Final answer: \(\boxed{-\frac{2}{3x+1}}\).
Explanation
- Analyze the given expression
The expression to simplify is \(\frac{-x-2}{9x^2-1} + \frac{-5x+4}{9x^2-1}\). Since both fractions share the same denominator, \(9x^2-1\), we can add their numerators directly according to the rules of addition for fractions with a common denominator: \(\frac{A}{D} + \frac{B}{D} = \frac{A+B}{D}\).
- Combine the numerators
We combine the numerators: \((-x-2) + (-5x+4) = -x - 5x - 2 + 4 = -6x + 2\). Thus, the combined expression is \(\frac{-6x+2}{9x^2-1}\).
- Factor the numerator and denominator
Now we factor both the numerator and the denominator. The numerator is \(-6x+2\), which can be factored as \(-2(3x-1)\). The denominator is \(9x^2-1\), which is a difference of squares \( (3x)^2 - (1)^2 \), so it factors as \((3x-1)(3x+1)\). Substituting these back into our expression, we have: \(\frac{-2(3x-1)}{(3x-1)(3x+1)}\).
- Simplify the expression and compare with the options
Given that \(x
eq \frac{1}{3}\) and \(x
eq -\frac{1}{3}\), we can divide both the numerator and the denominator by their common factor, \((3x-1)\). This yields \(\frac{-2}{3x+1}\), or equivalently, \(-\frac{2}{3x+1}\). Comparing this result with the given options, we identify the second option as the correct one.
Examples
Simplifying algebraic expressions like this is essential when dealing with rational functions in physics or engineering problems, such as analyzing the impedance of electrical circuits or calculating fluid flow rates through combined pipes. By reducing complex expressions to their simplest form, you make calculations easier and more intuitive to interpret, ensuring that you can identify relationships between different variables more efficiently in your models.