Algebra worksheet focusing on adding and subtracting rational expressions, designed for practice in simplifying complex fractions with variables.

Let's solve each of these adding and subtracting rational expressions problems step by step. The key idea is to:

1. Find a common denominator (if needed),
2. Rewrite each expression with the common denominator,
3. Combine the numerators,
4. Simplify the resulting expression (factor and cancel if possible).

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1) $\frac{u - v}{8v} + \frac{6u - 3v}{8v}$

✔ Same denominator: $8v$

Add numerators:
$$
\frac{(u - v) + (6u - 3v)}{8v} = \frac{u - v + 6u - 3v}{8v} = \frac{7u - 4v}{8v}
$$

✔ Answer: $\boxed{\frac{7u - 4v}{8v}}$

---

2) $\frac{m - 3n}{6m^3n} - \frac{m + 3n}{6m^3n}$

Same denominator: $6m^3n$

Subtract numerators:
$$
\frac{(m - 3n) - (m + 3n)}{6m^3n} = \frac{m - 3n - m - 3n}{6m^3n} = \frac{-6n}{6m^3n}
$$

Simplify:
$$
\frac{-6n}{6m^3n} = \frac{-1}{m^3}
$$

✔ Answer: $\boxed{-\frac{1}{m^3}}$

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3) $\frac{5}{a^2 + 3a + 2} + \frac{5a + 1}{a^2 + 3a + 2}$

Same denominator: factor it first.

$a^2 + 3a + 2 = (a+1)(a+2)$

Now add:
$$
\frac{5 + (5a + 1)}{(a+1)(a+2)} = \frac{5a + 6}{(a+1)(a+2)}
$$

No further simplification.

✔ Answer: $\boxed{\frac{5a + 6}{(a+1)(a+2)}}$

---

4) $\frac{5}{10n^2 + 16n + 6} + \frac{n - 6}{10n^2 + 16n + 6}$

Same denominator. Factor denominator:

$10n^2 + 16n + 6 = 2(5n^2 + 8n + 3)$

Try factoring: $5n^2 + 8n + 3 = (5n + 3)(n + 1)$

So denominator: $2(5n + 3)(n + 1)$

Now combine numerators:
$$
\frac{5 + (n - 6)}{10n^2 + 16n + 6} = \frac{n - 1}{10n^2 + 16n + 6}
$$

We can write:
$$
\frac{n - 1}{2(5n + 3)(n + 1)}
$$

No cancellation possible.

✔ Answer: $\boxed{\frac{n - 1}{10n^2 + 16n + 6}}$ or $\boxed{\frac{n - 1}{2(5n + 3)(n + 1)}}$

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5) $\frac{r + 6}{3r - 6} + \frac{r + 1}{3r - 6}$

Same denominator: $3r - 6 = 3(r - 2)$

Add numerators:
$$
\frac{(r + 6) + (r + 1)}{3r - 6} = \frac{2r + 7}{3(r - 2)}
$$

✔ Answer: $\boxed{\frac{2r + 7}{3(r - 2)}}$

---

6) $\frac{x + 2}{2x^2 + 13x + 20} - \frac{x + 3}{2x^2 + 13x + 20}$

Same denominator.

Factor: $2x^2 + 13x + 20 = (2x + 5)(x + 4)$

Subtract numerators:
$$
\frac{(x + 2) - (x + 3)}{(2x + 5)(x + 4)} = \frac{x + 2 - x - 3}{(2x + 5)(x + 4)} = \frac{-1}{(2x + 5)(x + 4)}
$$

✔ Answer: $\boxed{-\frac{1}{(2x + 5)(x + 4)}}$

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7) $\frac{6}{x - 1} - \frac{5x}{4}$

Different denominators: $x - 1$ and $4$

LCM: $4(x - 1)$

Rewrite both:
$$
\frac{6}{x - 1} = \frac{6 \cdot 4}{4(x - 1)} = \frac{24}{4(x - 1)}
$$
$$
\frac{5x}{4} = \frac{5x(x - 1)}{4(x - 1)} = \frac{5x^2 - 5x}{4(x - 1)}
$$

Now subtract:
$$
\frac{24 - (5x^2 - 5x)}{4(x - 1)} = \frac{24 - 5x^2 + 5x}{4(x - 1)} = \frac{-5x^2 + 5x + 24}{4(x - 1)}
$$

✔ Answer: $\boxed{\frac{-5x^2 + 5x + 24}{4(x - 1)}}$

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8) $6 - \frac{x + 5}{(7x - 5)(x + 4)}$

Write 6 as a fraction with denominator $(7x - 5)(x + 4)$:
$$
6 = \frac{6(7x - 5)(x + 4)}{(7x - 5)(x + 4)}
$$

Now subtract:
$$
\frac{6(7x - 5)(x + 4) - (x + 5)}{(7x - 5)(x + 4)}
$$

First compute $6(7x - 5)(x + 4)$

Step-by-step:
- $(7x - 5)(x + 4) = 7x(x) + 7x(4) -5(x) -5(4) = 7x^2 + 28x - 5x - 20 = 7x^2 + 23x - 20$
- Multiply by 6: $6(7x^2 + 23x - 20) = 42x^2 + 138x - 120$

Now subtract $(x + 5)$:
$$
42x^2 + 138x - 120 - x - 5 = 42x^2 + 137x - 125
$$

So:
$$
\frac{42x^2 + 137x - 125}{(7x - 5)(x + 4)}
$$

✔ Answer: $\boxed{\frac{42x^2 + 137x - 125}{(7x - 5)(x + 4)}}$

---

9) $\frac{3}{x + 7} + \frac{4}{x - 8}$

Different denominators: LCM = $(x + 7)(x - 8)$

Rewrite:
$$
\frac{3(x - 8)}{(x + 7)(x - 8)} + \frac{4(x + 7)}{(x + 7)(x - 8)} = \frac{3x - 24 + 4x + 28}{(x + 7)(x - 8)} = \frac{7x + 4}{(x + 7)(x - 8)}
$$

✔ Answer: $\boxed{\frac{7x + 4}{(x + 7)(x - 8)}}$

---

10) $\frac{3}{4v^2 + 4v} - \frac{7}{2}$

Factor denominator: $4v^2 + 4v = 4v(v + 1)$

LCM: $4v(v + 1)$

Rewrite:
$$
\frac{3}{4v(v + 1)} - \frac{7}{2} = \frac{3}{4v(v + 1)} - \frac{7 \cdot 2v(v + 1)}{4v(v + 1)} = \frac{3 - 14v(v + 1)}{4v(v + 1)}
$$

Compute numerator:
$$
14v(v + 1) = 14v^2 + 14v
$$
$$
3 - (14v^2 + 14v) = -14v^2 - 14v + 3
$$

✔ Answer: $\boxed{\frac{-14v^2 - 14v + 3}{4v(v + 1)}}$

---

11) $\frac{7}{3} - \frac{8}{12x - 8}$

Simplify second denominator: $12x - 8 = 4(3x - 2)$

So:
$$
\frac{7}{3} - \frac{8}{4(3x - 2)} = \frac{7}{3} - \frac{2}{3x - 2}
$$

LCM: $3(3x - 2)$

Rewrite:
$$
\frac{7(3x - 2)}{3(3x - 2)} - \frac{2 \cdot 3}{3(3x - 2)} = \frac{21x - 14 - 6}{3(3x - 2)} = \frac{21x - 20}{3(3x - 2)}
$$

✔ Answer: $\boxed{\frac{21x - 20}{3(3x - 2)}}$

---

12) $\frac{5}{n + 5} + \frac{4n}{2n + 6}$

Simplify second term: $2n + 6 = 2(n + 3)$

So:
$$
\frac{5}{n + 5} + \frac{4n}{2(n + 3)} = \frac{5}{n + 5} + \frac{2n}{n + 3}
$$

LCM: $(n + 5)(n + 3)$

Rewrite:
$$
\frac{5(n + 3)}{(n + 5)(n + 3)} + \frac{2n(n + 5)}{(n + 5)(n + 3)} = \frac{5n + 15 + 2n^2 + 10n}{(n + 5)(n + 3)} = \frac{2n^2 + 15n + 15}{(n + 5)(n + 3)}
$$

✔ Answer: $\boxed{\frac{2n^2 + 15n + 15}{(n + 5)(n + 3)}}$

---

13) $\frac{2x}{5x + 4} + \frac{6x}{2x + 3}$

Different denominators: LCM = $(5x + 4)(2x + 3)$

Rewrite:
$$
\frac{2x(2x + 3)}{(5x + 4)(2x + 3)} + \frac{6x(5x + 4)}{(5x + 4)(2x + 3)} = \frac{4x^2 + 6x + 30x^2 + 24x}{(5x + 4)(2x + 3)}
$$

Combine:
$$
(4x^2 + 30x^2) + (6x + 24x) = 34x^2 + 30x
$$

So:
$$
\frac{34x^2 + 30x}{(5x + 4)(2x + 3)} = \frac{2x(17x + 15)}{(5x + 4)(2x + 3)}
$$

✔ Answer: $\boxed{\frac{2x(17x + 15)}{(5x + 4)(2x + 3)}}$

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14) $\frac{2}{3x^2 + 12x} + \frac{8}{2x}$

Factor:
- $3x^2 + 12x = 3x(x + 4)$
- $2x = 2x$

LCM: $6x(x + 4)$

Rewrite:
$$
\frac{2}{3x(x + 4)} = \frac{2 \cdot 2}{6x(x + 4)} = \frac{4}{6x(x + 4)}
$$
$$
\frac{8}{2x} = \frac{8 \cdot 3(x + 4)}{6x(x + 4)} = \frac{24(x + 4)}{6x(x + 4)}
$$

Now add:
$$
\frac{4 + 24(x + 4)}{6x(x + 4)} = \frac{4 + 24x + 96}{6x(x + 4)} = \frac{24x + 100}{6x(x + 4)}
$$

Simplify:
Factor numerator: $4(6x + 25)$, denominator: $6x(x + 4)$

$$
= \frac{4(6x + 25)}{6x(x + 4)} = \frac{2(6x + 25)}{3x(x + 4)}
$$

✔ Answer: $\boxed{\frac{2(6x + 25)}{3x(x + 4)}}$

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $\frac{7u - 4v}{8v}$ |
| 2 | $-\frac{1}{m^3}$ |
| 3 | $\frac{5a + 6}{(a+1)(a+2)}$ |
| 4 | $\frac{n - 1}{10n^2 + 16n + 6}$ |
| 5 | $\frac{2r + 7}{3(r - 2)}$ |
| 6 | $-\frac{1}{(2x + 5)(x + 4)}$ |
| 7 | $\frac{-5x^2 + 5x + 24}{4(x - 1)}$ |
| 8 | $\frac{42x^2 + 137x - 125}{(7x - 5)(x + 4)}$ |
| 9 | $\frac{7x + 4}{(x + 7)(x - 8)}$ |
| 10 | $\frac{-14v^2 - 14v + 3}{4v(v + 1)}$ |
| 11 | $\frac{21x - 20}{3(3x - 2)}$ |
| 12 | $\frac{2n^2 + 15n + 15}{(n + 5)(n + 3)}$ |
| 13 | $\frac{2x(17x + 15)}{(5x + 4)(2x + 3)}$ |
| 14 | $\frac{2(6x + 25)}{3x(x + 4)}$ |

Let me know if you'd like these checked with a calculator or explained in more detail!