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).

You have 12 out of the total of 400 raffle tickets sold so you have a (\( \frac{12}{400} \)) x 100 = \( \frac{12 \times 100}{400} \) = \( \frac{1200}{400} \) = 3% chance to win the raffle.

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

\( \left|b - 1\right| \) - 2 = 5
\( \left|b - 1\right| \) = 5 + 2
\( \left|b - 1\right| \) = 7

The value inside the absolute value brackets can be either positive or negative so (b - 1) must equal + 7 or -7 for \( \left|b - 1\right| \) to equal 7:

So, b = -6 or b = 8.

Diane scored 77% on the test meaning she earned 77% of the possible points on the test. There were 120 possible points on the test so she earned 120 x 0.77 = 92 points. Each question is worth 4 points so she got \( \frac{92}{4} \) = 23 questions right.

In order to find how many additional workers are needed to staff the extra crews you first need to calculate how many workers are on a crew. There are 4 workers at the company now and that's enough to staff 2 crews so there are \( \frac{4}{2} \) = 2 workers on a crew. 4 crews are needed for the busy season which, at 2 workers per crew, means that the roofing company will need 4 x 2 = 8 total workers to staff the crews during the busy season. The company already employs 4 workers so they need to add 8 - 4 = 4 new staff for the busy season.