Free add and subtract rational expressions worksheet, Download ... - Free Printable

To solve the given problems involving adding and subtracting rational expressions, we need to follow a systematic approach. Here's how to simplify each expression step by step:

---

Problem 1:

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

Solution:
1. Both fractions have the same denominator, $8v$.
2. Add the numerators:
$$
(u - v) + (6u - 3v) = u - v + 6u - 3v = 7u - 4v
$$
3. Write the result as a single fraction:
$$
\frac{7u - 4v}{8v}
$$

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

---

Problem 2:

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

Solution:
1. Both fractions have the same denominator, $6m^3n$.
2. Subtract the numerators:
$$
(m - 3n) - (m + 3n) = m - 3n - m - 3n = -6n
$$
3. Write the result as a single fraction:
$$
\frac{-6n}{6m^3n}
$$
4. Simplify by canceling $6n$ from the numerator and denominator:
$$
\frac{-6n}{6m^3n} = \frac{-1}{m^3}
$$

Final Answer:
$$
\boxed{\frac{-1}{m^3}}
$$

---

Problem 3:

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

Solution:
1. Both fractions have the same denominator, $a^2 + 3a + 2$.
2. Add the numerators:
$$
5 + (5a + 1) = 5 + 5a + 1 = 5a + 6
$$
3. Write the result as a single fraction:
$$
\frac{5a + 6}{a^2 + 3a + 2}
$$
4. Factor the denominator:
$$
a^2 + 3a + 2 = (a + 1)(a + 2)
$$
5. The simplified expression is:
$$
\frac{5a + 6}{(a + 1)(a + 2)}
$$

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

---

Problem 4:

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

Solution:
1. Both fractions have the same denominator, $10n^2 + 16n + 6$.
2. Add the numerators:
$$
5 + (n - 6) = 5 + n - 6 = n - 1
$$
3. Write the result as a single fraction:
$$
\frac{n - 1}{10n^2 + 16n + 6}
$$
4. Factor the denominator:
$$
10n^2 + 16n + 6 = 2(5n^2 + 8n + 3) = 2(5n + 3)(n + 1)
$$
5. The simplified expression is:
$$
\frac{n - 1}{2(5n + 3)(n + 1)}
$$

Final Answer:
$$
\boxed{\frac{n - 1}{2(5n + 3)(n + 1)}}
$$

---

Problem 5:

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

Solution:
1. Both fractions have the same denominator, $3r - 6$.
2. Add the numerators:
$$
(r + 6) + (r + 1) = r + 6 + r + 1 = 2r + 7
$$
3. Write the result as a single fraction:
$$
\frac{2r + 7}{3r - 6}
$$
4. Factor the denominator:
$$
3r - 6 = 3(r - 2)
$$
5. The simplified expression is:
$$
\frac{2r + 7}{3(r - 2)}
$$

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

---

Problem 6:

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

Solution:
1. Both fractions have the same denominator, $2x^2 + 13x + 20$.
2. Subtract the numerators:
$$
(x + 2) - (x + 3) = x + 2 - x - 3 = -1
$$
3. Write the result as a single fraction:
$$
\frac{-1}{2x^2 + 13x + 20}
$$
4. Factor the denominator:
$$
2x^2 + 13x + 20 = (2x + 5)(x + 4)
$$
5. The simplified expression is:
$$
\frac{-1}{(2x + 5)(x + 4)}
$$

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

---

Problem 7:

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

Solution:
1. Find a common denominator. The denominators are $x - 1$ and $4$, so the least common denominator (LCD) is $4(x - 1)$.
2. Rewrite each fraction with the LCD:
$$
\frac{6}{x-1} = \frac{6 \cdot 4}{(x-1) \cdot 4} = \frac{24}{4(x-1)}
$$
$$
\frac{5x}{4} = \frac{5x \cdot (x-1)}{4 \cdot (x-1)} = \frac{5x(x-1)}{4(x-1)}
$$
3. Subtract the fractions:
$$
\frac{24}{4(x-1)} - \frac{5x(x-1)}{4(x-1)} = \frac{24 - 5x(x-1)}{4(x-1)}
$$
4. Simplify the numerator:
$$
24 - 5x(x-1) = 24 - 5x^2 + 5x = -5x^2 + 5x + 24
$$
5. Write the result:
$$
\frac{-5x^2 + 5x + 24}{4(x-1)}
$$

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

---

Problem 8:

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

Solution:
1. Rewrite $6$ as a fraction with the same denominator $(7x-5)(x+4)$:
$$
6 = \frac{6(7x-5)(x+4)}{(7x-5)(x+4)}
$$
2. Subtract the fractions:
$$
\frac{6(7x-5)(x+4)}{(7x-5)(x+4)} - \frac{x+5}{(7x-5)(x+4)} = \frac{6(7x-5)(x+4) - (x+5)}{(7x-5)(x+4)}
$$
3. Expand the numerator:
$$
6(7x-5)(x+4) = 6[(7x)(x) + (7x)(4) + (-5)(x) + (-5)(4)] = 6[7x^2 + 28x - 5x - 20] = 6[7x^2 + 23x - 20]
$$
$$
6(7x^2 + 23x - 20) = 42x^2 + 138x - 120
$$
4. Subtract $x + 5$ from the expanded numerator:
$$
42x^2 + 138x - 120 - (x + 5) = 42x^2 + 138x - 120 - x - 5 = 42x^2 + 137x - 125
$$
5. Write the result:
$$
\frac{42x^2 + 137x - 125}{(7x-5)(x+4)}
$$

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

---

Problem 9:

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

Solution:
1. Find a common denominator. The denominators are $x + 7$ and $x - 8$, so the LCD is $(x + 7)(x - 8)$.
2. Rewrite each fraction with the LCD:
$$
\frac{3}{x+7} = \frac{3(x-8)}{(x+7)(x-8)} = \frac{3x - 24}{(x+7)(x-8)}
$$
$$
\frac{4}{x-8} = \frac{4(x+7)}{(x-8)(x+7)} = \frac{4x + 28}{(x+7)(x-8)}
$$
3. Add the fractions:
$$
\frac{3x - 24}{(x+7)(x-8)} + \frac{4x + 28}{(x+7)(x-8)} = \frac{(3x - 24) + (4x + 28)}{(x+7)(x-8)}
$$
4. Simplify the numerator:
$$
(3x - 24) + (4x + 28) = 3x + 4x - 24 + 28 = 7x + 4
$$
5. Write the result:
$$
\frac{7x + 4}{(x+7)(x-8)}
$$

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

---

Problem 10:

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

Solution:
1. Factor the denominator of the first fraction:
$$
4v^2 + 4v = 4v(v + 1)
$$
2. Rewrite the second fraction with the same denominator $4v(v + 1)$:
$$
\frac{7}{2} = \frac{7 \cdot 2v(v+1)}{2 \cdot 2v(v+1)} = \frac{14v(v+1)}{4v(v+1)}
$$
3. Subtract the fractions:
$$
\frac{3}{4v(v+1)} - \frac{14v(v+1)}{4v(v+1)} = \frac{3 - 14v(v+1)}{4v(v+1)}
$$
4. Expand the numerator:
$$
3 - 14v(v+1) = 3 - 14v^2 - 14v
$$
5. Write the result:
$$
\frac{-14v^2 - 14v + 3}{4v(v+1)}
$$

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

---

Problem 11:

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

Solution:
1. Simplify the second fraction:
$$
\frac{8}{12x-8} = \frac{8}{4(3x-2)} = \frac{2}{3x-2}
$$
2. Rewrite $\frac{7}{3}$ with the same denominator $3(3x-2)$:
$$
\frac{7}{3} = \frac{7(3x-2)}{3(3x-2)} = \frac{21x - 14}{3(3x-2)}
$$
3. Rewrite $\frac{2}{3x-2}$ with the same denominator $3(3x-2)$:
$$
\frac{2}{3x-2} = \frac{2 \cdot 3}{(3x-2) \cdot 3} = \frac{6}{3(3x-2)}
$$
4. Subtract the fractions:
$$
\frac{21x - 14}{3(3x-2)} - \frac{6}{3(3x-2)} = \frac{(21x - 14) - 6}{3(3x-2)}
$$
5. Simplify the numerator:
$$
(21x - 14) - 6 = 21x - 14 - 6 = 21x - 20
$$
6. Write the result:
$$
\frac{21x - 20}{3(3x-2)}
$$

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

---

Problem 12:

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

Solution:
1. Simplify the second fraction:
$$
\frac{4n}{2n+6} = \frac{4n}{2(n+3)} = \frac{2n}{n+3}
$$
2. Find a common denominator. The denominators are $n+5$ and $n+3$, so the LCD is $(n+5)(n+3)$.
3. Rewrite each fraction with the LCD:
$$
\frac{5}{n+5} = \frac{5(n+3)}{(n+5)(n+3)} = \frac{5n + 15}{(n+5)(n+3)}
$$
$$
\frac{2n}{n+3} = \frac{2n(n+5)}{(n+3)(n+5)} = \frac{2n^2 + 10n}{(n+3)(n+5)}
$$
4. Add the fractions:
$$
\frac{5n + 15}{(n+5)(n+3)} + \frac{2n^2 + 10n}{(n+3)(n+5)} = \frac{(5n + 15) + (2n^2 + 10n)}{(n+5)(n+3)}
$$
5. Simplify the numerator:
$$
(5n + 15) + (2n^2 + 10n) = 2n^2 + 5n + 10n + 15 = 2n^2 + 15n + 15
$$
6. Write the result:
$$
\frac{2n^2 + 15n + 15}{(n+5)(n+3)}
$$

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

---

Problem 13:

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

Solution:
1. Find a common denominator. The denominators are $5x+4$ and $2x+3$, so the LCD is $(5x+4)(2x+3)$.
2. Rewrite each fraction with the LCD:
$$
\frac{2x}{5x+4} = \frac{2x(2x+3)}{(5x+4)(2x+3)} = \frac{4x^2 + 6x}{(5x+4)(2x+3)}
$$
$$
\frac{6x}{2x+3} = \frac{6x(5x+4)}{(2x+3)(5x+4)} = \frac{30x^2 + 24x}{(2x+3)(5x+4)}
$$
3. Add the fractions:
$$
\frac{4x^2 + 6x}{(5x+4)(2x+3)} + \frac{30x^2 + 24x}{(5x+4)(2x+3)} = \frac{(4x^2 + 6x) + (30x^2 + 24x)}{(5x+4)(2x+3)}
$$
4. Simplify the numerator:
$$
(4x^2 + 6x) + (30x^2 + 24x) = 4x^2 + 30x^2 + 6x + 24x = 34x^2 + 30x
$$
5. Write the result:
$$
\frac{34x^2 + 30x}{(5x+4)(2x+3)}
$$

Final Answer:
$$
\boxed{\frac{34x^2 + 30x}{(5x+4)(2x+3)}}
$$

---

Problem 14:

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

Solution:
1. Factor the denominator of the first fraction:
$$
3x^2 + 12x = 3x(x + 4)
$$
2. Rewrite the second fraction with the same denominator $3x(x+4)$:
$$
\frac{8}{2x} = \frac{8 \cdot 3(x+4)}{2x \cdot 3(x+4)} = \frac{24(x+4)}{6x(x+4)} = \frac{12(x+4)}{3x(x+4)}
$$
3. Add the fractions:
$$
\frac{2}{3x(x+4)} + \frac{12(x+4)}{3x(x+4)} = \frac{2 + 12(x+4)}{3x(x+4)}
$$
4. Simplify the numerator:
$$
2 + 12(x+4) = 2 + 12x + 48 = 12x + 50
$$
5. Write the result:
$$
\frac{12x + 50}{3x(x+4)}
$$

Final Answer:
$$
\boxed{\frac{12x + 50}{3x(x+4)}}
$$

---

Final Answers:

1. $\boxed{\frac{7u - 4v}{8v}}$
2. $\boxed{\frac{-1}{m^3}}$
3. $\boxed{\frac{5a + 6}{(a + 1)(a + 2)}}$
4. $\boxed{\frac{n - 1}{2(5n + 3)(n + 1)}}$
5. $\boxed{\frac{2r + 7}{3(r - 2)}}$
6. $\boxed{\frac{-1}{(2x + 5)(x + 4)}}$
7. $\boxed{\frac{-5x^2 + 5x + 24}{4(x-1)}}$
8. $\boxed{\frac{42x^2 + 137x - 125}{(7x-5)(x+4)}}$
9. $\boxed{\frac{7x + 4}{(x+7)(x-8)}}$
10. $\boxed{\frac{-14v^2 - 14v + 3}{4v(v+1)}}$
11. $\boxed{\frac{21x - 20}{3(3x-2)}}$
12. $\boxed{\frac{2n^2 + 15n + 15}{(n+5)(n+3)}}$
13. $\boxed{\frac{34x^2 + 30x}{(5x+4)(2x+3)}}$
14. $\boxed{\frac{12x + 50}{3x(x+4)}}$