Let’s revisit and solve the problems step-by-step for clarification and better readability.

1. Simplify

$\frac{2x + 1}{2} - \frac{3x - 7}{9} + \frac{5}{18}$

Step 1: Find the least common denominator (LCD) for $2$, $9$, and $18$, which is $18$.

Step 2: Rewrite each fraction with a denominator of $18$: $\frac{2x + 1}{2} = \frac{9(2x + 1)}{18}, \quad \frac{3x - 7}{9} = \frac{2(3x - 7)}{18}, \quad \frac{5}{18} = \frac{5}{18}.$

Step 3: Expand the numerators: $\frac{9(2x + 1)}{18} = \frac{18x + 9}{18}, \quad \frac{2(3x - 7)}{18} = \frac{6x - 14}{18}.$

Step 4: Combine all fractions: $\frac{18x + 9}{18} - \frac{6x - 14}{18} + \frac{5}{18}.$

Step 5: Combine numerators over the common denominator: $\frac{(18x + 9) - (6x - 14) + 5}{18} = \frac{18x + 9 - 6x + 14 + 5}{18}.$

Step 6: Simplify the numerator: $18x - 6x = 12x, \quad 9 + 14 + 5 = 28.$ $\frac{12x + 28}{18}.$

Step 7: Factorize and simplify: $\frac{12x + 28}{18} = \frac{4(3x + 7)}{6} = \frac{2(3x + 7)}{9}.$

Final Answer: $\frac{2(3x + 7)}{9}$.

2. Simplify

$\frac{x^2 - y^2}{xy + x^2} \times \frac{-2x^3}{xy - x^2}$

Step 1: Factorize:

• $x^2 - y^2 = (x - y)(x + y)$,
• $xy + x^2 = x(y + x)$,
• $xy - x^2 = x(y - x)$.

Step 2: Rewrite the expression: $\frac{(x - y)(x + y)}{x(y + x)} \times \frac{-2x^3}{x(y - x)}.$

Step 3: Combine the numerators and denominators: $\frac{(x - y)(x + y)(-2x^3)}{x(y + x)x(y - x)}.$

Step 4: Simplify: Cancel $x$ and common terms ($y + x$, etc.): $\frac{-2x^2(x + y)}{y - x}.$

Step 5: Simplify further by factoring out $-(x - y)$ from $y - x$: $\frac{-2x^2(x + y)}{-(x - y)} = \frac{2x^2(x + y)}{x - y}.$

Final Answer: $\frac{2x^2(x + y)}{x - y}$.

3. Simplify

$\frac{2}{2u + 3} \times \frac{3}{2u + 3} \div \frac{2}{4u^2 - 9}$

Step 1: Factorize $4u^2 - 9 = (2u + 3)(2u - 3)$.

Step 2: Rewrite the expression: $\frac{2}{2u + 3} \times \frac{3}{2u + 3} \div \frac{2}{(2u + 3)(2u - 3)}.$

Step 3: Division means multiplying by the reciprocal: $\frac{2}{2u + 3} \times \frac{3}{2u + 3} \times \frac{(2u + 3)(2u - 3)}{2}.$

Step 4: Simplify: $\frac{6(2u - 3)}{(2u + 3)^2}.$

Final Answer: $\frac{6(2u - 3)}{(2u + 3)^2}$.

4. Simplify

$\frac{x^2 - 9}{x^2 + 5x + 4} \times \frac{x^2 - 1}{x^2 + 6x + 9}$

Step 1: Factorize:

• $x^2 - 9 = (x - 3)(x + 3)$,
• $x^2 + 5x + 4 = (x + 4)(x + 1)$,
• $x^2 - 1 = (x - 1)(x + 1)$,
• $x^2 + 6x + 9 = (x + 3)(x + 3)$.

Step 2: Rewrite the expression: $\frac{(x - 3)(x + 3)}{(x + 4)(x + 1)} \times \frac{(x - 1)(x + 1)}{(x + 3)(x + 3)}.$

Step 3: Combine numerators and denominators: $\frac{(x - 3)(x + 3)(x - 1)(x + 1)}{(x + 4)(x + 1)(x + 3)(x + 3)}.$

Step 4: Cancel common terms:

• Cancel $x + 1$ and one $x + 3$: $\frac{(x - 3)(x - 1)}{(x + 4)(x + 3)}.$

Final Answer: $\frac{(x - 3)(x - 1)}{(x + 4)(x + 3)}$.

Do you have any specific questions or would you like further clarifications?

Related Questions

1. What is the importance of factoring in simplifying algebraic fractions?
2. How do you simplify division of fractions involving polynomials?
3. How does recognizing common terms help in simplifying expressions?
4. Can LCD methods apply to rational expressions with variables?
5. What steps ensure accuracy when simplifying complex fractions?

Tip: Always double-check your work for errors by substituting a simple value (e.g., $x = 1$) into the original and simplified expressions to see if they match.