Worksheet for practicing addition and subtraction of rational expressions with algebraic fractions.

Let’s solve each rational expression one by one. We’ll combine them by finding a common denominator, then simplifying the numerator.

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Problem 1:
\[
\frac{s}{3} - \frac{6s + 7}{s + 6}
\]

Step 1: Find common denominator → \(3(s + 6)\)

Step 2: Rewrite each fraction:
\[
\frac{s(s + 6)}{3(s + 6)} - \frac{3(6s + 7)}{3(s + 6)}
\]

Step 3: Combine numerators:
\[
\frac{s(s + 6) - 3(6s + 7)}{3(s + 6)} = \frac{s^2 + 6s - 18s - 21}{3(s + 6)} = \frac{s^2 - 12s - 21}{3(s + 6)}
\]

✔ Final for Problem 1: \(\frac{s^2 - 12s - 21}{3(s + 6)}\)

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Problem 2:
\[
\frac{2h}{6h + 8} + \frac{5}{7h + 4}
\]

Step 1: Factor denominators if possible.
- \(6h + 8 = 2(3h + 4)\)
- \(7h + 4\) is already simplified.

Common denominator: \(2(3h + 4)(7h + 4)\)

Step 2: Rewrite:
\[
\frac{2h \cdot (7h + 4)}{2(3h + 4)(7h + 4)} + \frac{5 \cdot 2(3h + 4)}{2(3h + 4)(7h + 4)}
\]

Step 3: Combine numerators:
Numerator = \(2h(7h + 4) + 10(3h + 4)\)
= \(14h^2 + 8h + 30h + 40 = 14h^2 + 38h + 40\)

So:
\[
\frac{14h^2 + 38h + 40}{2(3h + 4)(7h + 4)}
\]

We can factor numerator: 14h² + 38h + 40 → factor out 2: 2(7h² + 19h + 20)

Denominator has 2 → cancel:

\[
\frac{7h^2 + 19h + 20}{(3h + 4)(7h + 4)}
\]

✔ Final for Problem 2: \(\frac{7h^2 + 19h + 20}{(3h + 4)(7h + 4)}\)

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Problem 3:
\[
\frac{g}{g + 9} + \frac{8}{7g + 5}
\]

Common denominator: \((g + 9)(7g + 5)\)

Rewrite:
\[
\frac{g(7g + 5) + 8(g + 9)}{(g + 9)(7g + 5)}
\]

Numerator: \(7g^2 + 5g + 8g + 72 = 7g^2 + 13g + 72\)

✔ Final for Problem 3: \(\frac{7g^2 + 13g + 72}{(g + 9)(7g + 5)}\)

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Problem 4:
\[
\frac{8c}{3} - \frac{3c + 9}{6c + 7}
\]

Common denominator: \(3(6c + 7)\)

Rewrite:
\[
\frac{8c(6c + 7) - 3(3c + 9)}{3(6c + 7)}
\]

Numerator: \(48c^2 + 56c - 9c - 27 = 48c^2 + 47c - 27\)

✔ Final for Problem 4: \(\frac{48c^2 + 47c - 27}{3(6c + 7)}\)

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Problem 5:
\[
\frac{4b}{6b + 3} - \frac{6}{3b + 2}
\]

Factor first denominator: \(6b + 3 = 3(2b + 1)\)

Second denominator: \(3b + 2\) — no common factors with first.

Common denominator: \(3(2b + 1)(3b + 2)\)

Rewrite:
\[
\frac{4b \cdot (3b + 2) - 6 \cdot 3(2b + 1)}{3(2b + 1)(3b + 2)}
\]

Numerator: \(12b^2 + 8b - 18(2b + 1) = 12b^2 + 8b - 36b - 18 = 12b^2 - 28b - 18\)

Factor numerator: 2(6b² - 14b - 9)

Denominator: 3(2b+1)(3b+2)

No cancellation → leave as is.

✔ Final for Problem 5: \(\frac{12b^2 - 28b - 18}{3(2b + 1)(3b + 2)}\) or simplified: \(\frac{2(6b^2 - 14b - 9)}{3(2b + 1)(3b + 2)}\)

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Problem 6:
\[
\frac{g}{6} - \frac{6g + 2}{g + 3}
\]

Common denominator: \(6(g + 3)\)

Rewrite:
\[
\frac{g(g + 3) - 6(6g + 2)}{6(g + 3)}
\]

Numerator: \(g^2 + 3g - 36g - 12 = g^2 - 33g - 12\)

✔ Final for Problem 6: \(\frac{g^2 - 33g - 12}{6(g + 3)}\)

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Problem 7:
\[
\frac{4n}{6} - \frac{7n + 9}{6n + 2}
\]

Simplify first term: \(\frac{4n}{6} = \frac{2n}{3}\)

Second denominator: \(6n + 2 = 2(3n + 1)\)

Common denominator: \(3 \cdot 2(3n + 1) = 6(3n + 1)\)

Rewrite:
\[
\frac{2n \cdot 2(3n + 1) - 3(7n + 9)}{6(3n + 1)}
\]

Wait — better to use original forms to avoid confusion.

Original: \(\frac{4n}{6} - \frac{7n + 9}{6n + 2}\)

Common denominator: \(6(6n + 2)\)? No — better to simplify first.

Actually, let’s keep denominators as 6 and (6n+2). LCM of 6 and (6n+2).

Note: 6 = 2×3, 6n+2 = 2(3n+1) → LCM = 6(3n+1)

So rewrite:

First term: \(\frac{4n}{6} = \frac{4n(3n + 1)}{6(3n + 1)}\)

Second term: \(\frac{7n + 9}{6n + 2} = \frac{7n + 9}{2(3n + 1)} = \frac{3(7n + 9)}{6(3n + 1)}\)

Now subtract:

Numerator: \(4n(3n + 1) - 3(7n + 9) = 12n^2 + 4n - 21n - 27 = 12n^2 - 17n - 27\)

✔ Final for Problem 7: \(\frac{12n^2 - 17n - 27}{6(3n + 1)}\)

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Problem 8:
\[
\frac{2y}{3y + 8} - \frac{7}{4y + 4}
\]

Factor second denominator: \(4y + 4 = 4(y + 1)\)

First denominator: \(3y + 8\) — no common factors.

Common denominator: \(4(3y + 8)(y + 1)\)

Rewrite:
\[
\frac{2y \cdot 4(y + 1) - 7 \cdot (3y + 8)}{4(3y + 8)(y + 1)}
\]

Numerator: \(8y(y + 1) - 7(3y + 8) = 8y^2 + 8y - 21y - 56 = 8y^2 - 13y - 56\)

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

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Final 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. \(\frac{2(6b^2 - 14b - 9)}{3(2b + 1)(3b + 2)}\)
6. \(\frac{g^2 - 33g - 12}{6(g + 3)}\)
7. \(\frac{12n^2 - 17n - 27}{6(3n + 1)}\)
8. \(\frac{8y^2 - 13y - 56}{4(3y + 8)(y + 1)}\)

Final Answer:
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. \(\frac{2(6b^2 - 14b - 9)}{3(2b + 1)(3b + 2)}\)
6. \(\frac{g^2 - 33g - 12}{6(g + 3)}\)
7. \(\frac{12n^2 - 17n - 27}{6(3n + 1)}\)
8. \(\frac{8y^2 - 13y - 56}{4(3y + 8)(y + 1)}\)