Question

Simplify \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 \frac{x-3}{x+4} Step 4: Divide out common factors.

Factors and multiples
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.

2021-08-04
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 $$\displaystyle{\left({\frac{{{x}-{3}}}{{{x}+{4}}}}\right)}$$.