The absolute value is the positive magnitude of a particular number or variable and is indicated by two vertical lines: \(\left|-5\right| = 5\). In the case of a variable absolute value (\(\left|a\right| = 5\)) the value of a can be either positive or negative (a = -5 or a = 5).

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

\( \left|a - 3\right| \) + 3 = -9
\( \left|a - 3\right| \) = -9 - 3
\( \left|a - 3\right| \) = -12

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

So, a = 15 or a = -9.

Monica scored 82% on the test meaning she earned 82% of the possible points on the test. There were 100 possible points on the test so she earned 100 x 0.82 = 82 points. Each question is worth 2 points so she got \( \frac{82}{2} \) = 41 questions right.

Average speed in miles per hour is the number of miles traveled divided by the number of hours:

Solving for time:

time = \( \frac{\text{distance}}{\text{speed}} \)
time = \( \frac{520mi}{65mph} \)
8 hours

The yearly interest charged on this loan is the annual interest rate multiplied by the amount borrowed:

interest = annual interest rate x loan amount

i = (\( \frac{6}{100} \)) x $800
i = 0.06 x $800

No payments were made so the total amount due is the original amount + the accumulated interest: