How are both integers

Number Theory Level 5

The sum of absolute values of all real numbers $x$ , such that both of the fractions $\frac{x^2+4x-1}{7x^2-6x-5}$ and $\frac{1-x}{1+x}$ are integers, can be written as $\frac{a}{b}$ , where $a$ and $b$ are coprime positive integers. What is the value of $a+b$ ?

Details and assumptions

If you think that there are no real numbers $x$ , then the (empty) sum is 0, and you should type your numerical answer as 1.

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