Question

Simplify.
\(\displaystyle{\frac{{{1}+{\frac{{{2}}}{{{x}+{4}}}}}}{{{1}+{\frac{{{9}}}{{{x}-{3}}}}}}}\)
Step 1: Find the LCM of the denominators of the fractions in the numerator and denominator.
Step 2: Multiply the numerator and denominator of the complex fraction by the LCM.
Step 3: Factor \(\displaystyle{\frac{{{x}-{3}}}{{{x}+{4}}}}\)
Step 4: Divide out common factors.

Answer 1

Given fraction is:
\(\displaystyle{\frac{{{1}+{\frac{{{2}}}{{{x}+{4}}}}}}{{{1}+{\frac{{{9}}}{{{x}-{3}}}}}}}\)
To simplify the given fraction.
Step1. Find the LCM of the denominators of the fractions in numerators and denominators
\(\displaystyle{\frac{{{1}+{\frac{{{2}}}{{{x}+{4}}}}}}{{{1}+{\frac{{{9}}}{{{x}-{3}}}}}}}={\frac{{{\frac{{{\left({x}+{4}\right)}+{2}}}{{{\left({x}+{4}\right)}}}}}}{{{\frac{{{\left({x}-{3}\right)}+{9}}}{{{\left({x}-{3}\right)}}}}}}}\)
\(\displaystyle{\frac{{{\frac{{{\left({x}+{6}\right)}}}{{{\left({x}+{4}\right)}}}}}}{{{\frac{{{\left({x}+{6}\right)}}}{{{x}-{3}}}}}}}\)
Step 2. Multiply the numerator and denominator of the complex fraction by the LCM.
\(\displaystyle{\frac{{{\frac{{{\left({x}+{6}\right)}}}{{{\left({x}+{4}\right)}}}}}}{{{\frac{{{\left({x}+{6}\right)}}}{{{x}-{3}}}}}}}={\frac{{{\left({x}+{6}\right)}\times{\left({x}-{3}\right)}}}{{{\left({x}+{4}\right)}\times{\left({x}+{6}\right)}}}}\)
Step 3. Factor
\(\displaystyle{\frac{{{\left({x}+{6}\right)}\times{\left({x}-{3}\right)}}}{{{\left({x}+{4}\right)}\times{\left({x}+{6}\right)}}}}\)
Step 4. Divide out common factors
\(\displaystyle{\frac{{{\left({x}+{6}\right)}\times{\left({x}-{3}\right)}}}{{{\left({x}+{4}\right)}\times{\left({x}+{6}\right)}}}}={\left({\frac{{{x}-{3}}}{{{x}+{4}}}}\right)}\)
Thus, the simplified form is \(\displaystyle{\left({\frac{{{x}-{3}}}{{{x}+{4}}}}\right)}\).