Here are the solutions to the problems: 21. Reduce the following expression into a single fraction $$\frac{4x-5}{2} - \frac{2x-1}{6}$$ Step 1: Find a common denominator for the two fractions. The least common multiple of 2 and 6 is 6. Step 2: Rewrite the first fraction with the denominator 6. To do this, multiply both the numerator and the denominator by 3. $$\frac{3(4x-5)}{3 \times 2} - \frac{2x-1}{6}$$ $$\frac{12x-15}{6} - \frac{2x-1}{6}$$ Step 3: Combine the fractions by subtracting the numerators. Remember to distribute the negative sign to all terms in the second numerator. $$\frac{(12x-15) - (2x-1)}{6}$$ $$\frac{12x-15-2x+1}{6}$$ Step 4: Combine like terms in the numerator. $$\frac{(12x-2x) + (-15+1)}{6}$$ $$\frac{10x-14}{6}$$ Step 5: Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor, which is 2. $$\frac{2(5x-7)}{2 \times 3}$$ $$\boxed{\frac{5x-7}{3}}$$ 22. Given that $1.0\dot{5} = 1\frac{a}{b}$, Find the values of a and b. Step 1: Convert the repeating decimal $1.0\dot{5}$ into a fraction. Let $N = 1.0\dot{5}$ Multiply by 10 to move the non-repeating part to the left of the decimal: $$10N = 10.5\dot{5} \quad (1)$$ Multiply by 100 to move one full repeating block to the left of the decimal: $$100N = 105.\dot{5} \quad (2)$$ Step 2: Subtract equation (1) from equation (2) to eliminate the repeating part. $$100N - 10N = 105.\dot{5} - 10.5\dot{5}$$ $$90N = 95$$ Step 3: Solve for $N$. $$N = \frac{95}{90}$$ Step 4: Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 5. $$N = \frac{95 \div 5}{90 \div 5} = \frac{19}{18}$$ Step 5: Express the improper fraction as a mixed number. $$\frac{19}{18} = 1\frac{1}{18}$$ Step 6: Compare this to the given form $1\frac{a}{b}$. 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. Step 1: Convert the repeating decimal $2.\dot{6}$ into a fraction. Let $M = 2.\dot{6}$ $$10M = 26.\dot{6}$$ Subtract $M$ from $10M$: $$10M - M = 26.\dot{6} - 2.\dot{6}$$ $$9M = 24$$ $$M = \frac{24}{9}$$ Simplify the fraction by dividing by 3: $$M = \frac{8}{3}$$ So, $x = \frac{8}{3}$. Step 2: Evaluate the numerator of the fraction. Numerator $= -2^2 - (-8)$ Recall that $-2^2 = -(2 \times 2) = -4$. Numerator $= -4 - (-8)$ Numerator $= -4 + 8$ Numerator $= 4$ Step 3: Evaluate the denominator of the fraction, substituting $x = \frac{8}{3}$. 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 $= \frac{-28}{3}$ Step 4: Evaluate the entire fraction. $$\frac{\text{Numerator}}{\text{Denominator}} = \frac{4}{\frac{-28}{3}}$$ To divide by a fraction, multiply by its reciprocal: $$4 \times \left(\frac{3}{-28}\right)$$ $$= \frac{12}{-28}$$ Step 5: Simplify the fraction. Divide both the numerator and the denominator by their greatest common divisor, which is 4. $$= -\frac{12 \div 4}{28 \div 4}$$ $$= \boxed{-\frac{3}{7}}$$