Reduce the following expression into a single fraction

The least common multiple of 2 and 6 is 6.

To do this, multiply both the numerator and the denominator by 3.

Remember to distribute the negative sign to all terms in the second numerator.

22. Given that $1.0\dot{5} = 1\frac{a}{b}$, Find the values of a and b.

Let $N = 1.0\dot{5}$
Multiply by 10 to move the non-repeating part to the left of the decimal:

Multiply by 100 to move one full repeating block to the left of the decimal:

By comparison, $a=1$ and $b=18$.
The values are $\boxed{a=1, b=18}$.

23. Evaluate $\frac{-2^2 - (-8)}{-2x - 12 \div 3}$ of $2.\dot{6}$ without using a calculator, in simplest form.
The phrase "Evaluate [expression] of [value]" implies substituting the value into the expression for the variable $x$. So, we will substitute $x = 2.\dot{6}$ into the given fraction.

Let $M = 2.\dot{6}$

Subtract $M$ from $10M$:

Simplify the fraction by dividing by 3:

So, $x = \frac{8}{3}$.

Numerator $= -2^2 - (-8)$
Recall that $-2^2 = -(2 \times 2) = -4$.
Numerator $= -4 - (-8)$
Numerator $= -4 + 8$
Numerator $= 4$

Denominator $= -2x - 12 \div 3$
First, perform the division: $12 \div 3 = 4$.
Denominator $= -2x - 4$
Now substitute $x = \frac{8}{3}$:
Denominator $= -2\left(\frac{8}{3}\right) - 4$
Denominator $= -\frac{16}{3} - 4$
To subtract, find a common denominator for $-\frac{16}{3}$ and 4. $4 = \frac{12}{3}$.
Denominator $= -\frac{16}{3} - \frac{12}{3}$
Denominator $= \frac{-16 - 12}{3}$
Denominator $= \frac{-28}{3}$

To divide by a fraction, multiply by its reciprocal:

Divide both the numerator and the denominator by their greatest common divisor, which is 4.