Add.
$\frac{-x-2}{9 x^2-1}+\frac{-5 x+4}{9 x^2-1}$
We are given the sum of two fractions with the same denominator:
\[\frac{-x-2}{9x^2-1} + \frac{-5x+4}{9x^2-1}.\]
Our objective is to combine these fractions and simplify the resulting expression. The denominator is \(9x^2-1\), which is a difference of squares, and the numerators contain linear terms in \(x\).
Since the fractions have the same denominator, we can add the numerators directly:
\[\frac{(-x-2) + (-5x+4)}{9x^2-1}.\]
Now, we combine like terms in the numerator:
\[(-x - 5x) + (-2 + 4) = -6x + 2.\]
So the combined fraction is:
\[\frac{-6x+2}{9x^2-1}.\]
To simplify the expression \(\frac{-6x+2}{9x^2-1}\), we factor both the numerator and the denominator.
For the numerator, we factor out -2:
\[-6x+2 = -2(3x-1).\]
For the denominator, we use the difference of squares formula \(a^2 - b^2 = (a-b)(a+b)\):
\[9x^2-1 = (3x)^2 - 1^2 = (3x-1)(3x+1).\]
Now substitute these back into the fraction:
\[\frac{-2(3x-1)}{(3x-1)(3x+1)}.\]
Assuming \(x
eq \frac{1}{3}\), we can cancel the common factor \((3x-1)\) from the numerator and the denominator:
\[\frac{-2(3x-1)}{(3x-1)(3x+1)} = \frac{-2}{3x+1}.\]
Thus, the simplified form of the expression is \(\frac{-2}{3x+1}\).
Simplifying rational expressions is an essential skill in algebra that allows us to manage complex formulas in physics, engineering, and economics. For example, when analyzing the output of a system described by a rational function, we might need to add or subtract multiple components to find the overall effect. By simplifying the result, we can more easily interpret the behavior of the system, such as determining its vertical asymptotes or horizontal limits, which represent critical points in real-world scenarios.