Question

1. \\(\dfrac{\left(\dfrac{1}{a - b} - \dfrac{3}{a + b}\

ight)}{\left(\dfrac{2}{b - a} + \dfrac{4}{b + a}\
ight)}\\)

Explanation:

Step1: Simplify numerator of the main fraction

First, simplify the numerator \(\frac{1}{a - b}-\frac{3}{a + b}\). Note that \(b - a=-(a - b)\), but in the numerator, we have \(a - b\) and \(a + b\). Find a common denominator \((a - b)(a + b)\):

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Step2: Simplify denominator of the main fraction

Now, simplify the denominator \(\frac{2}{b - a}+\frac{4}{b + a}\). Since \(b - a=-(a - b)\), we can rewrite \(\frac{2}{b - a}\) as \(-\frac{2}{a - b}\). Then find a common denominator \((a - b)(a + b)\) (or \((b - a)(b + a)\), which is the same as \((a - b)(a + b)\) with a negative sign, but we can use \((a - b)(a + b)\)):

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Step3: Divide the simplified numerator by the simplified denominator

Now, the original expression is the division of the simplified numerator by the simplified denominator:

$$ \frac{\frac{-2a + 4b}{(a - b)(a + b)}}{\frac{2a-6b}{(a - b)(a + b)}} $$

When dividing fractions, we multiply by the reciprocal:

$$ \frac{-2a + 4b}{(a - b)(a + b)}\times\frac{(a - b)(a + b)}{2a-6b} $$

The \((a - b)(a + b)\) terms cancel out:

$$ \frac{-2a + 4b}{2a-6b} $$

Factor out \(-2\) from the numerator and \(2\) from the denominator:

$$ \frac{-2(a - 2b)}{2(a - 3b)} $$

Cancel out the common factor of \(2\):

$$ \frac{-(a - 2b)}{a - 3b}=\frac{-a + 2b}{a - 3b}=\frac{2b - a}{a - 3b} $$

Answer:

\(\frac{2b - a}{a - 3b}\) (or \(\frac{-a + 2b}{a - 3b}\))