Question

perform the indicated operation and simplify completely. \\(\frac{x + 5}{x + 7} + \frac{6 - 4x}{x - 9}\\) question help: \\(\boxed{\text{video}}\\)

Explanation:

Step1: Identify common denominator

The denominators are $x+7$ and $x-9$, so the common denominator is $(x+7)(x-9)$.

Step2: Rewrite fractions with common denominator

$$\frac{(x+5)(x-9)}{(x+7)(x-9)} + \frac{(6-4x)(x+7)}{(x+7)(x-9)}$$

Step3: Expand numerators

First numerator: $(x+5)(x-9)=x^2-9x+5x-45=x^2-4x-45$
Second numerator: $(6-4x)(x+7)=6x+42-4x^2-28x=-4x^2-22x+42$

Step4: Add numerators

$$(x^2-4x-45)+(-4x^2-22x+42)=x^2-4x^2-4x-22x-45+42=-3x^2-26x-3$$

Step5: Combine into single fraction

$$\frac{-3x^2-26x-3}{(x+7)(x-9)}$$

Step6: Factor numerator (if possible)

The numerator $-3x^2-26x-3$ cannot be factored over integers, so we can factor out a negative sign:
$$-\frac{3x^2+26x+3}{(x+7)(x-9)}$$

Answer:

$\displaystyle -\frac{3x^2+26x+3}{(x+7)(x-9)}$ (or equivalent $\displaystyle \frac{-3x^2-26x-3}{(x+7)(x-9)}$)