First, solve for \( \left|b + 5\right| \):

\( \left|b + 5\right| \) - 7 = 6
\( \left|b + 5\right| \) = 6 + 7
\( \left|b + 5\right| \) = 13

The value inside the absolute value brackets can be either positive or negative so (b + 5) must equal + 13 or -13 for \( \left|b + 5\right| \) to equal 13:

So, b = -18 or b = 8.

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 2 are [2, 4, 6, 8, 10, 12, 14, 16, 18, 20] and the first few multiples of 10 are [10, 20, 30, 40, 50, 60, 70, 80, 90]. The first few multiples they share are [10, 20, 30, 40, 50] making 10 the smallest multiple 2 and 10 share.

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

\( \frac{4 x 5}{2 x 5} \) - \( \frac{9 x 1}{10 x 1} \)

\( \frac{20}{10} \) - \( \frac{9}{10} \)

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

\( \frac{20 - 9}{10} \) = \( \frac{11}{10} \) = 1\(\frac{1}{10}\)

A rational number (or fraction) is represented as a ratio between two integers, a and b, and has the form \({a \over b}\) where a is the numerator and b is the denominator. An improper fraction (\({5 \over 3} \)) has a numerator with a greater absolute value than the denominator and can be converted into a mixed number (\(1 {2 \over 3} \)) which has a whole number part and a fractional part.

The least common multiple (LCM) is the smallest positive integer that is a multiple of two or more integers.

The hourly error rate for this machine is the error rate in parts per 100 multiplied by the number of parts produced per hour:

\( \frac{6}{100} \) x 8 = \( \frac{6 \times 8}{100} \) = \( \frac{48}{100} \) = 0.48 errors per hour

So, in an average hour, the machine will produce 8 - 0.48 = 7.52 error free parts.

The machine ran for 24 - 6 = 18 hours yesterday so you would expect that 18 x 7.52 = 135.4 error free parts were produced yesterday.