An integer is any whole number, including zero. An integer can be either positive or negative. Examples include -77, -1, 0, 55, 119.

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 6 are [6, 12, 18, 24, 30, 36, 42, 48, 54, 60] and the first few multiples of 14 are [14, 28, 42, 56, 70, 84, 98]. The first few multiples they share are [42, 84] making 42 the smallest multiple 6 and 14 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{4 x 7}{6 x 7} \) + \( \frac{8 x 3}{14 x 3} \)

\( \frac{28}{42} \) + \( \frac{24}{42} \)

Now, because the fractions share a common denominator, you can add them:

\( \frac{28 + 24}{42} \) = \( \frac{52}{42} \) = 1\(\frac{5}{21}\)

First, solve for \( \left|c - 1\right| \):

\( \left|c - 1\right| \) + 3 = -8
\( \left|c - 1\right| \) = -8 - 3
\( \left|c - 1\right| \) = -11

The value inside the absolute value brackets can be either positive or negative so (c - 1) must equal - 11 or --11 for \( \left|c - 1\right| \) to equal -11:

So, c = 12 or c = -10.

A factorial is the product of an integer and all the positive integers below it. To solve a fraction featuring factorials, expand the factorials and cancel out like numbers:

\( \frac{5!}{6!} \)
\( \frac{5 \times 4 \times 3 \times 2 \times 1}{6 \times 5 \times 4 \times 3 \times 2 \times 1} \)
\( \frac{1}{6} \)
\( \frac{1}{6} \)

The equation for this sequence is:

a_n = a_n-1 + 5

where n is the term's order in the sequence, a_n is the value of the term, and a_n-1 is the value of the term before a_n. This makes the next number:

a₆ = a₅ + 5
a₆ = 21 + 5
a₆ = 26