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:

\( \frac{2!}{4!} \)
\( \frac{2 \times 1}{4 \times 3 \times 2 \times 1} \)
\( \frac{1}{4 \times 3} \)
\( \frac{1}{12} \)

31% of the water consumption is industrial use and 15% is personal use so (31% - 15%) = 16% more water is used for industrial purposes. 25,000 gallons are consumed daily so industry consumes \( \frac{16}{100} \) x 25,000 gallons = 4,000 gallons.

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 3 are [3, 6, 9, 12, 15, 18, 21, 24, 27, 30] and the first few multiples of 5 are [5, 10, 15, 20, 25, 30, 35, 40, 45, 50]. The first few multiples they share are [15, 30, 45, 60, 75] making 15 the smallest multiple 3 and 5 share.

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

\( \frac{7 x 5}{3 x 5} \) - \( \frac{8 x 3}{5 x 3} \)

\( \frac{35}{15} \) - \( \frac{24}{15} \)

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

\( \frac{35 - 24}{15} \) = \( \frac{11}{15} \) = \(\frac{11}{15}\)

The distributive property for multiplication helps in solving expressions like a(b + c). It specifies that the result of multiplying one number by the sum or difference of two numbers can be obtained by multiplying each number individually and then totaling the results: a(b + c) = ab + ac. For example, 4(10-5) = (4 x 10) - (4 x 5) = 40 - 20 = 20.

First, solve for \( \left|c - 9\right| \):

\( \left|c - 9\right| \) - 9 = 5
\( \left|c - 9\right| \) = 5 + 9
\( \left|c - 9\right| \) = 14

The value inside the absolute value brackets can be either positive or negative so (c - 9) must equal + 14 or -14 for \( \left|c - 9\right| \) to equal 14:

So, c = -5 or c = 23.