Adding and Subtracting Rational Expressions Worksheets - Math Monks - Free Printable

Let's solve each problem on the Rational Expression Worksheet step by step. We'll focus on adding and subtracting rational expressions, which involves finding a common denominator, combining numerators, and simplifying.

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Problem 1:

$$
\frac{7}{y - 5} - \frac{4}{y - 5}
$$

✔ Same denominator, so subtract numerators:
$$
= \frac{7 - 4}{y - 5} = \frac{3}{y - 5}
$$

✔ Answer: $\boxed{\frac{3}{y - 5}}$

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Problem 2:

$$
\frac{4}{a + 1} + \frac{2}{a + 1}
$$

✔ Same denominator:
$$
= \frac{4 + 2}{a + 1} = \frac{6}{a + 1}
$$

✔ Answer: $\boxed{\frac{6}{a + 1}}$

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Problem 3:

$$
\frac{5}{3x^2} - \frac{1}{3x^2}
$$

✔ Same denominator:
$$
= \frac{5 - 1}{3x^2} = \frac{4}{3x^2}
$$

✔ Answer: $\boxed{\frac{4}{3x^2}}$

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Problem 4:

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

✘ Different denominators: $x - 1$ and $4$.
Find LCD: $4(x - 1)$

Convert both fractions:
- $\frac{6}{x - 1} = \frac{6 \cdot 4}{4(x - 1)} = \frac{24}{4(x - 1)}$
- $\frac{5x}{4} = \frac{5x(x - 1)}{4(x - 1)} = \frac{5x^2 - 5x}{4(x - 1)}$

Now subtract:
$$
\frac{24 - (5x^2 - 5x)}{4(x - 1)} = \frac{24 - 5x^2 + 5x}{4(x - 1)}
$$

Rewriting:
$$
= \frac{-5x^2 + 5x + 24}{4(x - 1)}
$$

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

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Problem 5:

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

First, simplify second denominator:
$$
2x + 6 = 2(x + 3)
$$

So:
$$
\frac{5}{x + 5} + \frac{4x}{2(x + 3)} = \frac{5}{x + 5} + \frac{2x}{x + 3}
$$

Now, different denominators: $x + 5$ and $x + 3$.
LCD: $(x + 5)(x + 3)$

Convert:
- $\frac{5}{x + 5} = \frac{5(x + 3)}{(x + 5)(x + 3)} = \frac{5x + 15}{(x + 5)(x + 3)}$
- $\frac{2x}{x + 3} = \frac{2x(x + 5)}{(x + 3)(x + 5)} = \frac{2x^2 + 10x}{(x + 5)(x + 3)}$

Add:
$$
\frac{5x + 15 + 2x^2 + 10x}{(x + 5)(x + 3)} = \frac{2x^2 + 15x + 15}{(x + 5)(x + 3)}
$$

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

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Problem 6:

$$
\frac{6}{x - 5} + \frac{x + 2}{x - 5}
$$

✔ Same denominator:
$$
= \frac{6 + (x + 2)}{x - 5} = \frac{x + 8}{x - 5}
$$

✔ Answer: $\boxed{\frac{x + 8}{x - 5}}$

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Problem 7:

$$
\frac{4}{5b} - \frac{7a}{4b^2}
$$

Different denominators: $5b$ and $4b^2$

LCD: $20b^2$

Convert:
- $\frac{4}{5b} = \frac{4 \cdot 4b}{20b^2} = \frac{16b}{20b^2}$
- $\frac{7a}{4b^2} = \frac{7a \cdot 5}{20b^2} = \frac{35a}{20b^2}$

Subtract:
$$
\frac{16b - 35a}{20b^2}
$$

✔ Answer: $\boxed{\frac{16b - 35a}{20b^2}}$

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Problem 8:

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

Denominators: $5x + 4$ and $2x + 3$ — no common factors.

LCD: $(5x + 4)(2x + 3)$

Convert:
- $\frac{2x}{5x + 4} = \frac{2x(2x + 3)}{(5x + 4)(2x + 3)} = \frac{4x^2 + 6x}{(5x + 4)(2x + 3)}$
- $\frac{6x}{2x + 3} = \frac{6x(5x + 4)}{(2x + 3)(5x + 4)} = \frac{30x^2 + 24x}{(5x + 4)(2x + 3)}$

Add:
$$
\frac{4x^2 + 6x + 30x^2 + 24x}{(5x + 4)(2x + 3)} = \frac{34x^2 + 30x}{(5x + 4)(2x + 3)}
$$

Factor numerator:
$$
= \frac{2x(17x + 15)}{(5x + 4)(2x + 3)}
$$

✔ Answer: $\boxed{\frac{2x(17x + 15)}{(5x + 4)(2x + 3)}}$

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1 | $\frac{3}{y - 5}$ |
| 2 | $\frac{6}{a + 1}$ |
| 3 | $\frac{4}{3x^2}$ |
| 4 | $\frac{-5x^2 + 5x + 24}{4(x - 1)}$ |
| 5 | $\frac{2x^2 + 15x + 15}{(x + 5)(x + 3)}$ |
| 6 | $\frac{x + 8}{x - 5}$ |
| 7 | $\frac{16b - 35a}{20b^2}$ |
| 8 | $\frac{2x(17x + 15)}{(5x + 4)(2x + 3)}$ |

Let me know if you'd like these simplified further or checked for domain restrictions!