Math worksheet for adding and subtracting rational expressions with algebraic fractions.

Here are the step-by-step solutions for each problem on the worksheet.

Left Column

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

* Step 1: Find the common denominator, which is $3(s + 6)$.
* Step 2: Multiply the top and bottom of the first fraction by $(s + 6)$ and the second fraction by $3$.
$$ \frac{s(s + 6)}{3(s + 6)} - \frac{3(6s + 7)}{3(s + 6)} $$
* Step 3: Expand the numerators. Be careful to distribute the negative sign to the second part.
$$ \frac{(s^2 + 6s) - (18s + 21)}{3(s + 6)} $$
$$ \frac{s^2 + 6s - 18s - 21}{3(s + 6)} $$
* Step 4: Combine like terms ($6s - 18s = -12s$).
$$ \frac{s^2 - 12s - 21}{3(s + 6)} $$

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

* Step 1: The denominators don't share factors, so the common denominator is $(g + 9)(7g + 5)$.
* Step 2: Cross-multiply the numerators.
$$ \frac{g(7g + 5) + 8(g + 9)}{(g + 9)(7g + 5)} $$
* Step 3: Expand and simplify the numerator.
$$ \frac{7g^2 + 5g + 8g + 72}{(g + 9)(7g + 5)} $$
$$ \frac{7g^2 + 13g + 72}{(g + 9)(7g + 5)} $$

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

* Step 1: Factor the first denominator: $6b + 3 = 3(2b + 1)$. The expression becomes:
$$ \frac{4b}{3(2b + 1)} - \frac{6}{3b + 2} $$
* Step 2: The common denominator is $3(2b + 1)(3b + 2)$.
* Step 3: Adjust the fractions.
$$ \frac{4b(3b + 2) - 6[3(2b + 1)]}{3(2b + 1)(3b + 2)} $$
* Step 4: Expand the numerator.
$$ \frac{(12b^2 + 8b) - (36b + 18)}{3(2b + 1)(3b + 2)} $$
$$ \frac{12b^2 + 8b - 36b - 18}{3(2b + 1)(3b + 2)} $$
* Step 5: Combine like terms.
$$ \frac{12b^2 - 28b - 18}{3(2b + 1)(3b + 2)} $$
*(Note: You can factor out a 2 from the top, but it doesn't cancel with the 3 on the bottom, so this form is correct.)*

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

* Step 1: Simplify the first fraction: $\frac{4n}{6} = \frac{2n}{3}$. Factor the second denominator: $6n + 2 = 2(3n + 1)$.
$$ \frac{2n}{3} - \frac{7n + 9}{2(3n + 1)} $$
* Step 2: Common denominator is $6(3n + 1)$.
* Step 3: Adjust fractions.
$$ \frac{2n[2(3n + 1)] - 3(7n + 9)}{6(3n + 1)} $$
* Step 4: Expand numerator.
$$ \frac{4n(3n + 1) - (21n + 27)}{6(3n + 1)} $$
$$ \frac{12n^2 + 4n - 21n - 27}{6(3n + 1)} $$
* Step 5: Combine like terms.
$$ \frac{12n^2 - 17n - 27}{6(3n + 1)} $$

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Right Column

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

* Step 1: Factor the first denominator: $6h + 8 = 2(3h + 4)$.
$$ \frac{2h}{2(3h + 4)} + \frac{5}{7h + 4} $$
* Step 2: Simplify the first fraction by canceling the 2s: $\frac{h}{3h + 4}$.
$$ \frac{h}{3h + 4} + \frac{5}{7h + 4} $$
* Step 3: Common denominator is $(3h + 4)(7h + 4)$.
* Step 4: Cross-multiply numerators.
$$ \frac{h(7h + 4) + 5(3h + 4)}{(3h + 4)(7h + 4)} $$
* Step 5: Expand and combine.
$$ \frac{7h^2 + 4h + 15h + 20}{(3h + 4)(7h + 4)} $$
$$ \frac{7h^2 + 19h + 20}{(3h + 4)(7h + 4)} $$

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

* Step 1: Common denominator is $3(6c + 7)$.
* Step 2: Adjust fractions.
$$ \frac{8c(6c + 7) - 3(3c + 9)}{3(6c + 7)} $$
* Step 3: Expand numerator.
$$ \frac{(48c^2 + 56c) - (9c + 27)}{3(6c + 7)} $$
$$ \frac{48c^2 + 56c - 9c - 27}{3(6c + 7)} $$
* Step 4: Combine like terms.
$$ \frac{48c^2 + 47c - 27}{3(6c + 7)} $$

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

* Step 1: Common denominator is $6(g + 3)$.
* Step 2: Adjust fractions.
$$ \frac{g(g + 3) - 6(6g + 2)}{6(g + 3)} $$
* Step 3: Expand numerator.
$$ \frac{(g^2 + 3g) - (36g + 12)}{6(g + 3)} $$
$$ \frac{g^2 + 3g - 36g - 12}{6(g + 3)} $$
* Step 4: Combine like terms.
$$ \frac{g^2 - 33g - 12}{6(g + 3)} $$

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

* Step 1: Factor the second denominator: $4y + 4 = 4(y + 1)$.
$$ \frac{2y}{3y + 8} - \frac{7}{4(y + 1)} $$
* Step 2: Common denominator is $4(3y + 8)(y + 1)$.
* Step 3: Adjust fractions.
$$ \frac{2y[4(y + 1)] - 7(3y + 8)}{4(3y + 8)(y + 1)} $$
* Step 4: Expand numerator.
$$ \frac{8y(y + 1) - (21y + 56)}{4(3y + 8)(y + 1)} $$
$$ \frac{8y^2 + 8y - 21y - 56}{4(3y + 8)(y + 1)} $$
* Step 5: Combine like terms.
$$ \frac{8y^2 - 13y - 56}{4(3y + 8)(y + 1)} $$

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