Question

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Explanation:

First Problem: Divide and Simplify $\boldsymbol{\frac{x + 3}{2} \div \frac{x - 2}{x}}$

Step1: Recall division of fractions rule

To divide two fractions, we multiply the first fraction by the reciprocal of the second fraction. So, $\frac{x + 3}{2} \div \frac{x - 2}{x}=\frac{x + 3}{2}\times\frac{x}{x - 2}$.

Step2: Multiply the numerators and denominators

Multiply the numerators: $(x + 3)\times x=x(x + 3)$ and the denominators: $2\times(x - 2) = 2(x - 2)$. So the simplified form is $\frac{x(x + 3)}{2(x - 2)}$.

Step1: Subtract the numerators (same denominator)

Since the denominators are the same, we subtract the numerators: $(6 - 4n)-(2 - 5n)=6 - 4n - 2 + 5n$.

Step2: Simplify the numerator

Simplify $6 - 4n - 2 + 5n$: Combine like terms, $6-2=4$ and $-4n + 5n=n$, so the numerator becomes $n + 4$. The fraction is now $\frac{n + 4}{n(n + 4)}$.

Step3: Cancel out common factors

Cancel out the common factor $(n + 4)$ from the numerator and the denominator (note that $n
eq - 4$ so we can cancel). We get $\frac{1}{n}$.

Answer:

$\frac{x(x + 3)}{2(x - 2)}$

Second Problem: Combine and Simplify $\boldsymbol{\frac{6 - 4n}{n(n + 4)}-\frac{2 - 5n}{n(n + 4)}}$