B: $$ \frac{x+6}{2 x-6} \div \frac{5 x+30}{6 x-18} $$

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UpStudy AI · Accepted Answer

We start with 
$$
B = \frac{x+6}{2x-6} \div \frac{5x+30}{6x-18}.
$$
First, rewrite the division as multiplication by the reciprocal:
$$
B = \frac{x+6}{2x-6} \times \frac{6x-18}{5x+30}.
$$

Next, factor common factors in each polynomial:
- In $$2x-6$$, factor out $$2$$:
  $$
  2x-6 = 2(x-3).
  $$
- In $$6x-18$$, factor out $$6$$:
  $$
  6x-18 = 6(x-3).
  $$
- In $$5x+30$$, factor out $$5$$:
  $$
  5x+30 = 5(x+6).
  $$

Substitute these factors back into the expression:
$$
B = \frac{x+6}{2(x-3)} \times \frac{6(x-3)}{5(x+6)}.
$$

Now, cancel the common factors:
- Cancel $$x+6$$ from the numerator of the first fraction and the denominator of the second fraction (provided $$x \neq -6$$).
- Cancel $$(x-3)$$ from the denominator of the first fraction and the numerator of the second fraction (provided $$x \neq 3$$).

This leaves:
$$
B = \frac{6}{2 \cdot 5}.
$$

Simplify the fraction:
$$
B = \frac{6}{10} = \frac{3}{5}.
$$

Thus, the simplified expression is:
$$
\frac{3}{5}.
$$
The simplified expression is $$ \frac{3}{5} $$.

B: \( \frac{x+6}{2 x-6} \div \frac{5 x+30}{6 x-18} \)

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