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

Step 1: Identify the denominators: $3$ and $s + 6$.
The least common denominator (LCD) is $3(s + 6)$.

Step 2: Rewrite each fraction with the LCD:

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

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

Step 3: Subtract:

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

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

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

Step 1: Factor denominators:

- $6h + 8 = 2(3h + 4)$
- $7h + 4$ is already simplified.

So denominators are $2(3h + 4)$ and $7h + 4$. No common factors → LCD is $2(3h + 4)(7h + 4)$

Step 2: Rewrite both fractions:

$$
\frac{2h}{2(3h + 4)} = \frac{2h(7h + 4)}{2(3h + 4)(7h + 4)} = \frac{2h(7h + 4)}{2(3h + 4)(7h + 4)}
$$

$$
\frac{5}{7h + 4} = \frac{5 \cdot 2(3h + 4)}{2(3h + 4)(7h + 4)} = \frac{10(3h + 4)}{2(3h + 4)(7h + 4)}
$$

Step 3: Add numerators:

$$
\frac{2h(7h + 4) + 10(3h + 4)}{2(3h + 4)(7h + 4)}
= \frac{14h^2 + 8h + 30h + 40}{2(3h + 4)(7h + 4)}
= \frac{14h^2 + 38h + 40}{2(3h + 4)(7h + 4)}
$$

Step 4: Simplify numerator and denominator:

Factor numerator: $14h^2 + 38h + 40$

Try factoring:
$$
= 2(7h^2 + 19h + 20)
$$

Check discriminant: $19^2 - 4(7)(20) = 361 - 560 = -199$ → not factorable over integers.

So leave as is.

Now divide numerator and denominator by 2:

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

✔ 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: $g + 9$, $7g + 5$ — no common factors.

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

Rewrite:

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

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

Add:

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

Check if numerator factors: $7g^2 + 13g + 72$

Discriminant: $13^2 - 4(7)(72) = 169 - 2016 = -1847$ → no real roots.

✔ 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: $3$, $6c + 7$

LCD: $3(6c + 7)$

Rewrite:

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

$$
\frac{3c + 9}{6c + 7} = \frac{3(3c + 9)}{3(6c + 7)} = \frac{9c + 27}{3(6c + 7)}
$$

Subtract:

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

Check if numerator factors: $48c^2 + 47c - 27$

Use AC method: $AC = 48 \times (-27) = -1296$

Find two numbers that multiply to $-1296$ and add to $47$: Try $72$ and $-18$: $72 - 18 = 54$ → no.

Try $81$ and $-16$: $81 - 16 = 65$ → no.

Try $108$ and $-12$: $108 - 12 = 96$ → too big.

Try $54$ and $-24$: $54 - 24 = 30$ → no.

No obvious factorization. So leave as is.

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

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

First simplify: $6b + 3 = 3(2b + 1)$

So:
$$
\frac{4b}{3(2b + 1)} - \frac{6}{3b + 2}
$$

Denominators: $3(2b + 1)$, $3b + 2$ → no common factors

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

Rewrite:

$$
\frac{4b}{3(2b + 1)} = \frac{4b(3b + 2)}{3(2b + 1)(3b + 2)} = \frac{12b^2 + 8b}{3(2b + 1)(3b + 2)}
$$

$$
\frac{6}{3b + 2} = \frac{6 \cdot 3(2b + 1)}{3(2b + 1)(3b + 2)} = \frac{18(2b + 1)}{3(2b + 1)(3b + 2)} = \frac{36b + 18}{3(2b + 1)(3b + 2)}
$$

Subtract:

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

Simplify numerator: factor out 2:

$$
= \frac{2(6b^2 - 14b - 9)}{3(2b + 1)(3b + 2)}
$$

Check if $6b^2 - 14b - 9$ factors: AC = $-54$

Look for two numbers that multiply to $-54$, add to $-14$: $-18$ and $+4$: $-18 + 4 = -14$ → yes!

So:
$$
6b^2 - 14b - 9 = 6b^2 - 18b + 4b - 9 = 6b(b - 3) + 1(4b - 9) → doesn't work.

Try grouping: $6b^2 - 18b + 4b - 9 = 6b(b - 3) + 1(4b - 9)$ → no.

Try: $6b^2 - 14b - 9 = (3b + 1)(2b - 9)?$

$(3b + 1)(2b - 9) = 6b^2 - 27b + 2b - 9 = 6b^2 - 25b - 9$ → no.

Try $(3b - 9)(2b + 1) = 6b^2 + 3b - 18b - 9 = 6b^2 - 15b - 9$ → close.

No clean factorization.

So leave as is.

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

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

Denominators: $6$, $g + 3$

LCD: $6(g + 3)$

Rewrite:

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

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

Subtract:

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

✔ 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:

$\frac{4n}{6} = \frac{2n}{3}$

Denominator: $6n + 2 = 2(3n + 1)$

So:
$$
\frac{2n}{3} - \frac{7n + 9}{2(3n + 1)}
$$

LCD: $6(3n + 1)$

Rewrite:

$$
\frac{2n}{3} = \frac{2n \cdot 2(3n + 1)}{6(3n + 1)} = \frac{4n(3n + 1)}{6(3n + 1)} = \frac{12n^2 + 4n}{6(3n + 1)}
$$

$$
\frac{7n + 9}{2(3n + 1)} = \frac{3(7n + 9)}{6(3n + 1)} = \frac{21n + 27}{6(3n + 1)}
$$

Subtract:

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

Check if numerator factors: $12n^2 - 17n - 27$

AC = $12 \times (-27) = -324$

Find two numbers that multiply to $-324$, add to $-17$: try $-27$ and $12$: $-27 + 12 = -15$ → no

Try $-36$ and $9$: $-36 + 9 = -27$

Try $-27$ and $12$: no

Try $-18$ and $18$: no

Try $-27$ and $12$: no

Try $-27$ and $12$: no

Try $-36$ and $9$: $-36 + 9 = -27$

Try $-27$ and $12$: $-27 + 12 = -15$

Try $-24$ and $13.5$: not integer

No easy factorization.

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

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

Simplify second denominator: $4y + 4 = 4(y + 1)$

So:
$$
\frac{2y}{3y + 8} - \frac{7}{4(y + 1)}
$$

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

Rewrite:

$$
\frac{2y}{3y + 8} = \frac{2y \cdot 4(y + 1)}{4(3y + 8)(y + 1)} = \frac{8y(y + 1)}{4(3y + 8)(y + 1)} = \frac{8y^2 + 8y}{4(3y + 8)(y + 1)}
$$

$$
\frac{7}{4(y + 1)} = \frac{7(3y + 8)}{4(3y + 8)(y + 1)} = \frac{21y + 56}{4(3y + 8)(y + 1)}
$$

Subtract:

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

Check if numerator factors: $8y^2 - 13y - 56$

AC = $8 \times (-56) = -448$

Find two numbers that multiply to $-448$, add to $-13$

Try $-32$ and $14$: $-32 + 14 = -18$

Try $-28$ and $16$: $-28 + 16 = -12$

Try $-22$ and $20$: $-22 + 20 = -2$

Try $-35$ and $12$: $-35 + 12 = -23$

Try $-16$ and $28$: $-16 + 28 = 12$

Try $-21$ and $16$: $-21 + 16 = -5$

Try $-24$ and $19$: no

Not easily factorable.

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

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✔ Final Answers Summary:

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 further or simplified differently!