Add & Subtract Rational Expressions Worksheets (printable, online ... - Free Printable

Let's solve each of the rational expression problems step by step. We will add or subtract the given rational expressions by finding a common denominator, combining the numerators, and simplifying where possible.

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1.

$$
\frac{s}{3} - \frac{6s + 7}{s + 6}
$$

- The denominators are $3$ and $s + 6$. The least common denominator (LCD) is $3(s + 6)$.
- Rewrite each fraction with the LCD:

$$
\frac{s}{3} = \frac{s(s + 6)}{3(s + 6)} = \frac{s^2 + 6s}{3(s + 6)}
$$
$$
\frac{6s + 7}{s + 6} = \frac{3(6s + 7)}{3(s + 6)} = \frac{18s + 21}{3(s + 6)}
$$

Now subtract:
$$
\frac{s^2 + 6s - (18s + 21)}{3(s + 6)} = \frac{s^2 + 6s - 18s - 21}{3(s + 6)} = \frac{s^2 - 12s - 21}{3(s + 6)}
$$

✔ Final Answer:
$$
\boxed{\frac{s^2 - 12s - 21}{3(s + 6)}}
$$

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2.

$$
\frac{2h}{6h + 8} + \frac{5}{7h + 4}
$$

First, factor denominators:
- $6h + 8 = 2(3h + 4)$
- $7h + 4$ is already simplified.

So the LCD is $2(3h + 4)(7h + 4)$

Rewrite each fraction:

$$
\frac{2h}{2(3h + 4)} = \frac{h}{3h + 4}
$$

Now we have:
$$
\frac{h}{3h + 4} + \frac{5}{7h + 4}
$$

LCD is $(3h + 4)(7h + 4)$

$$
= \frac{h(7h + 4)}{(3h + 4)(7h + 4)} + \frac{5(3h + 4)}{(3h + 4)(7h + 4)}
$$
$$
= \frac{7h^2 + 4h + 15h + 20}{(3h + 4)(7h + 4)} = \frac{7h^2 + 19h + 20}{(3h + 4)(7h + 4)}
$$

✔ Final Answer:
$$
\boxed{\frac{7h^2 + 19h + 20}{(3h + 4)(7h + 4)}}
$$

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3.

$$
\frac{g}{g + 9} + \frac{8}{7g + 5}
$$

Denominators: $g + 9$ and $7g + 5$ — no common factors.

LCD = $(g + 9)(7g + 5)$

$$
= \frac{g(7g + 5)}{(g + 9)(7g + 5)} + \frac{8(g + 9)}{(g + 9)(7g + 5)}
$$
$$
= \frac{7g^2 + 5g + 8g + 72}{(g + 9)(7g + 5)} = \frac{7g^2 + 13g + 72}{(g + 9)(7g + 5)}
$$

✔ Final Answer:
$$
\boxed{\frac{7g^2 + 13g + 72}{(g + 9)(7g + 5)}}
$$

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4.

$$
\frac{8c}{3} - \frac{3c + 9}{6c + 7}
$$

Denominators: $3$ and $6c + 7$

LCD = $3(6c + 7)$

$$
\frac{8c}{3} = \frac{8c(6c + 7)}{3(6c + 7)} = \frac{48c^2 + 56c}{3(6c + 7)}
$$
$$
\frac{3c + 9}{6c + 7} = \frac{3(3c + 9)}{3(6c + 7)} = \frac{9c + 27}{3(6c + 7)}
$$

Subtract:
$$
\frac{48c^2 + 56c - (9c + 27)}{3(6c + 7)} = \frac{48c^2 + 47c - 27}{3(6c + 7)}
$$

✔ Final Answer:
$$
\boxed{\frac{48c^2 + 47c - 27}{3(6c + 7)}}
$$

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5.

$$
\frac{4b}{6b + 3} - \frac{6}{3b + 2}
$$

Factor:
- $6b + 3 = 3(2b + 1)$
- $3b + 2$ stays as is

LCD = $3(2b + 1)(3b + 2)$

Rewrite:
$$
\frac{4b}{3(2b + 1)} = \frac{4b(3b + 2)}{3(2b + 1)(3b + 2)} = \frac{12b^2 + 8b}{3(2b + 1)(3b + 2)}
$$
$$
\frac{6}{3b + 2} = \frac{6 \cdot 3(2b + 1)}{3(2b + 1)(3b + 2)} = \frac{18(2b + 1)}{3(2b + 1)(3b + 2)} = \frac{36b + 18}{3(2b + 1)(3b + 2)}
$$

Now subtract:
$$
\frac{12b^2 + 8b - (36b + 18)}{3(2b + 1)(3b + 2)} = \frac{12b^2 - 28b - 18}{3(2b + 1)(3b + 2)}
$$

We can factor numerator:
Try factoring $12b^2 - 28b - 18$

Divide all terms by 2: $6b^2 - 14b - 9$, doesn't factor nicely.

So leave as is.

✔ Final Answer:
$$
\boxed{\frac{12b^2 - 28b - 18}{3(2b + 1)(3b + 2)}}
$$

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6.

$$
\frac{g}{6} - \frac{6g + 2}{g + 3}
$$

LCD = $6(g + 3)$

$$
\frac{g}{6} = \frac{g(g + 3)}{6(g + 3)} = \frac{g^2 + 3g}{6(g + 3)}
$$
$$
\frac{6g + 2}{g + 3} = \frac{6(6g + 2)}{6(g + 3)} = \frac{36g + 12}{6(g + 3)}
$$

Subtract:
$$
\frac{g^2 + 3g - (36g + 12)}{6(g + 3)} = \frac{g^2 - 33g - 12}{6(g + 3)}
$$

✔ Final Answer:
$$
\boxed{\frac{g^2 - 33g - 12}{6(g + 3)}}
$$

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7.

$$
\frac{4n}{6} - \frac{7n + 9}{6n + 2}
$$

Simplify first:
- $\frac{4n}{6} = \frac{2n}{3}$

Denominators: $3$ and $6n + 2 = 2(3n + 1)$

LCD = $6(3n + 1)$

Now:
$$
\frac{2n}{3} = \frac{2n \cdot 2(3n + 1)}{6(3n + 1)} = \frac{4n(3n + 1)}{6(3n + 1)} = \frac{12n^2 + 4n}{6(3n + 1)}
$$
$$
\frac{7n + 9}{6n + 2} = \frac{7n + 9}{2(3n + 1)} = \frac{3(7n + 9)}{6(3n + 1)} = \frac{21n + 27}{6(3n + 1)}
$$

Subtract:
$$
\frac{12n^2 + 4n - (21n + 27)}{6(3n + 1)} = \frac{12n^2 - 17n - 27}{6(3n + 1)}
$$

✔ Final Answer:
$$
\boxed{\frac{12n^2 - 17n - 27}{6(3n + 1)}}
$$

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8.

$$
\frac{2y}{3y + 8} - \frac{7}{4y + 4}
$$

Factor:
- $4y + 4 = 4(y + 1)$

LCD = $4(3y + 8)(y + 1)$

Rewrite:
$$
\frac{2y}{3y + 8} = \frac{2y \cdot 4(y + 1)}{4(3y + 8)(y + 1)} = \frac{8y(y + 1)}{4(3y + 8)(y + 1)} = \frac{8y^2 + 8y}{4(3y + 8)(y + 1)}
$$
$$
\frac{7}{4y + 4} = \frac{7}{4(y + 1)} = \frac{7(3y + 8)}{4(3y + 8)(y + 1)} = \frac{21y + 56}{4(3y + 8)(y + 1)}
$$

Subtract:
$$
\frac{8y^2 + 8y - (21y + 56)}{4(3y + 8)(y + 1)} = \frac{8y^2 - 13y - 56}{4(3y + 8)(y + 1)}
$$

✔ Final Answer:
$$
\boxed{\frac{8y^2 - 13y - 56}{4(3y + 8)(y + 1)}}
$$

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✔ Summary of All Answers:

1. $\frac{s^2 - 12s - 21}{3(s + 6)}$
2. $\frac{7h^2 + 19h + 20}{(3h + 4)(7h + 4)}$
3. $\frac{7g^2 + 13g + 72}{(g + 9)(7g + 5)}$
4. $\frac{48c^2 + 47c - 27}{3(6c + 7)}$
5. $\frac{12b^2 - 28b - 18}{3(2b + 1)(3b + 2)}$
6. $\frac{g^2 - 33g - 12}{6(g + 3)}$
7. $\frac{12n^2 - 17n - 27}{6(3n + 1)}$
8. $\frac{8y^2 - 13y - 56}{4(3y + 8)(y + 1)}$

Let me know if you'd like any of these checked for further simplification!