Question

divide as indicated
\\(\frac{x + 1}{7} \div \frac{5x + 5}{6}\\)
select the correct choice below and fill in the answer box(es) to complete your choice.
(simplify your answer. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)
a. \\(\frac{x + 1}{7} \div \frac{5x + 5}{6} = \square, x \
eq \square\\)
b. \\(\frac{x + 1}{7} \div \frac{5x + 5}{6} = \square\\), no numbers must be excluded.

Explanation:

Step1: Rewrite division as multiplication

$\frac{x+1}{7} \div \frac{5x+5}{6} = \frac{x+1}{7} \times \frac{6}{5x+5}$

Step2: Factor denominator of second term

$5x+5 = 5(x+1)$, so expression becomes $\frac{x+1}{7} \times \frac{6}{5(x+1)}$

Step3: Cancel common factors

Cancel $(x+1)$ from numerator and denominator: $\frac{1}{7} \times \frac{6}{5}$

Step4: Multiply remaining fractions

$\frac{1 \times 6}{7 \times 5} = \frac{6}{35}$

Step5: Identify excluded values

Original expression is undefined when denominators are 0 or when the divisor is 0:

• $7

eq 0$ (always true)

• $5x+5=0 \implies x=-1$
• $x+1=0 \implies x=-1$

So $x
eq -1$

Answer:

A. $\frac{x+1}{7} \div \frac{5x+5}{6} = \frac{6}{35}$, $x
eq -1$