First, solve for \( \left|c - 3\right| \):

\( \left|c - 3\right| \) - 6 = 6
\( \left|c - 3\right| \) = 6 + 6
\( \left|c - 3\right| \) = 12

The value inside the absolute value brackets can be either positive or negative so (c - 3) must equal + 12 or -12 for \( \left|c - 3\right| \) to equal 12:

So, c = -9 or c = 15.

The area of a circle is given by the formula A = πr² where r is the radius of the circle. The radius of a circle is its diameter divided by two so A = π(\( \frac{d}{2} \))². If the diameter of the logo increases by 35% the radius (and, consequently, the total area) increases by \( \frac{35\text{%}}{2} \) = 17\(\frac{1}{2}\)%

To take the square root of a fraction, break the fraction into two separate roots then calculate the square root of the numerator and denominator separately:

\( \sqrt{\frac{4}{9}} \)
\( \frac{\sqrt{4}}{\sqrt{9}} \)
\( \frac{\sqrt{2^2}}{\sqrt{3^2}} \)
\(\frac{2}{3}\)

To add these distances, they must share the same unit so first you need to first convert the swim distance from meters (m) to kilometers (km) before adding it to the bike and run distances which are already in km. To convert 100 meters to kilometers, divide the distance by 1000 to get 0.1km then add the remaining distances:

total distance = swim + bike + run
total distance = 0.1km + 30.4km + 12.3km
total distance = 42.8km

The distributive property for division helps in solving expressions like \({b + c \over a}\). It specifies that the result of dividing a fraction with multiple terms in the numerator and one term in the denominator can be obtained by dividing each term individually and then totaling the results: \({b + c \over a} = {b \over a} + {c \over a}\). For example, \({a^3 + 6a^2 \over a^2} = {a^3 \over a^2} + {6a^2 \over a^2} = a + 6\).