This algebra worksheet provides practice problems for simplifying rational expressions using multiplication, division, and addition.

Let's solve each problem step-by-step, starting with Section A and then moving to Section B, simplifying rational expressions where possible.

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🔷 Section A: Simplify the following

We'll simplify each expression using algebraic rules for multiplication and division of fractions.

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#### 1) $\frac{ac}{b} \times \frac{b}{c}$

Multiply numerators and denominators:
$$
\frac{ac \cdot b}{b \cdot c} = \frac{abc}{bc}
$$
Cancel $b$ and $c$:
$$
= a
$$

✔ Answer: $a$

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#### 2) $\frac{a + b}{3ab} \times \frac{6a^2}{2a + 2b}$

Factor denominator: $2a + 2b = 2(a + b)$
$$
= \frac{a + b}{3ab} \times \frac{6a^2}{2(a + b)}
$$
Cancel $(a + b)$ from numerator and denominator:
$$
= \frac{1}{3ab} \times \frac{6a^2}{2} = \frac{1}{3ab} \times 3a^2 = \frac{3a^2}{3ab} = \frac{a}{b}
$$

✔ Answer: $\frac{a}{b}$

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#### 3) $\frac{x^2}{x^2 - 2x} \times \frac{3 + x}{x}$

Factor:
- $x^2 - 2x = x(x - 2)$
- $3 + x = x + 3$

So:
$$
= \frac{x^2}{x(x - 2)} \times \frac{x + 3}{x}
$$
Simplify:
$$
= \frac{x}{x - 2} \times \frac{x + 3}{x} = \frac{x(x + 3)}{(x - 2)x} = \frac{x + 3}{x - 2}
$$

✔ Answer: $\frac{x + 3}{x - 2}$

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#### 4) $\frac{(3x + 2)^2}{6x} \times \frac{x^4}{6x + 4}$

Note: $6x + 4 = 2(3x + 2)$

So:
$$
= \frac{(3x + 2)^2}{6x} \times \frac{x^4}{2(3x + 2)} = \frac{(3x + 2) \cdot (3x + 2)}{6x} \times \frac{x^4}{2(3x + 2)}
$$
Cancel one $(3x + 2)$:
$$
= \frac{3x + 2}{6x} \times \frac{x^4}{2} = \frac{(3x + 2)x^4}{12x}
$$
Simplify $x^4 / x = x^3$:
$$
= \frac{(3x + 2)x^3}{12}
$$

✔ Answer: $\frac{(3x + 2)x^3}{12}$

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#### 5) $\frac{x^2 y^2}{y} \div \frac{x^4 y}{x}$

First simplify each fraction:

- $\frac{x^2 y^2}{y} = x^2 y$
- $\frac{x^4 y}{x} = x^3 y$

Now divide:
$$
x^2 y \div x^3 y = \frac{x^2 y}{x^3 y} = \frac{1}{x}
$$

✔ Answer: $\frac{1}{x}$

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#### 6) $\frac{b - 3}{2b^2 - 5b - 3} \div (2b - 6)$

First factor denominator:
- $2b^2 - 5b - 3$: Find two numbers that multiply to $2 \cdot (-3) = -6$, add to $-5$: $-6$ and $1$
- So: $2b^2 - 6b + b - 3 = 2b(b - 3) + 1(b - 3) = (2b + 1)(b - 3)$

So:
$$
\frac{b - 3}{(2b + 1)(b - 3)} \div (2b - 6)
$$
Cancel $b - 3$:
$$
= \frac{1}{2b + 1} \div (2b - 6)
$$
Now $2b - 6 = 2(b - 3)$, so:
$$
= \frac{1}{2b + 1} \times \frac{1}{2(b - 3)} = \frac{1}{2(2b + 1)(b - 3)}
$$

✔ Answer: $\frac{1}{2(2b + 1)(b - 3)}$

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#### 7) $\frac{1}{y^2 + 3y + 2} \div \frac{2}{y^2 - 4}$

Factor:
- $y^2 + 3y + 2 = (y + 1)(y + 2)$
- $y^2 - 4 = (y - 2)(y + 2)$

So:
$$
\frac{1}{(y + 1)(y + 2)} \div \frac{2}{(y - 2)(y + 2)} = \frac{1}{(y + 1)(y + 2)} \times \frac{(y - 2)(y + 2)}{2}
$$
Cancel $(y + 2)$:
$$
= \frac{1}{y + 1} \times \frac{y - 2}{2} = \frac{y - 2}{2(y + 1)}
$$

✔ Answer: $\frac{y - 2}{2(y + 1)}$

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#### 8) $\frac{a^2(a - 2)}{y^2} \div \frac{a(a - 2)}{y^5}$

Divide by multiplying by reciprocal:
$$
= \frac{a^2(a - 2)}{y^2} \times \frac{y^5}{a(a - 2)}
$$
Cancel:
- $a^2 / a = a$
- $(a - 2)$ cancels
- $y^5 / y^2 = y^3$

So:
$$
= a \cdot y^3 = a y^3
$$

✔ Answer: $a y^3$

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✔ Section A Summary:

| Problem | Answer |
|--------|--------|
| 1 | $a$ |
| 2 | $\frac{a}{b}$ |
| 3 | $\frac{x + 3}{x - 2}$ |
| 4 | $\frac{(3x + 2)x^3}{12}$ |
| 5 | $\frac{1}{x}$ |
| 6 | $\frac{1}{2(2b + 1)(b - 3)}$ |
| 7 | $\frac{y - 2}{2(y + 1)}$ |
| 8 | $a y^3$ |

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🔷 Section B: Write as single fractions

We will combine terms into single fractions using common denominators.

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#### 1) $\frac{x}{5} + 4$

Write 4 as $\frac{20}{5}$:
$$
= \frac{x + 20}{5}
$$

✔ Answer: $\frac{x + 20}{5}$

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#### 2) $y - \frac{6}{y}$

Write $y = \frac{y^2}{y}$:
$$
= \frac{y^2 - 6}{y}
$$

✔ Answer: $\frac{y^2 - 6}{y}$

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#### 3) $4 - \frac{5}{x - 2}$

Write $4 = \frac{4(x - 2)}{x - 2} = \frac{4x - 8}{x - 2}$:
$$
= \frac{4x - 8 - 5}{x - 2} = \frac{4x - 13}{x - 2}
$$

✔ Answer: $\frac{4x - 13}{x - 2}$

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#### 4) $\frac{1}{3w + 5} + 9$

Write $9 = \frac{9(3w + 5)}{3w + 5}$:
$$
= \frac{1 + 9(3w + 5)}{3w + 5} = \frac{1 + 27w + 45}{3w + 5} = \frac{27w + 46}{3w + 5}
$$

✔ Answer: $\frac{27w + 46}{3w + 5}$

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#### 5) $6 + \frac{3}{x + 2} + x$

Group: $(6 + x) + \frac{3}{x + 2} = (x + 6) + \frac{3}{x + 2}$

Write $x + 6 = \frac{(x + 6)(x + 2)}{x + 2}$:
$$
= \frac{(x + 6)(x + 2) + 3}{x + 2}
$$
Expand:
$$
(x + 6)(x + 2) = x^2 + 2x + 6x + 12 = x^2 + 8x + 12
$$
Add 3:
$$
= \frac{x^2 + 8x + 15}{x + 2}
$$

✔ Answer: $\frac{x^2 + 8x + 15}{x + 2}$

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#### 6) $\frac{3}{x} + \frac{5}{x} = \frac{8}{x}$

✔ Answer: $\frac{8}{x}$

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#### 7) $\frac{7a}{4} + \frac{11a}{4} = \frac{18a}{4} = \frac{9a}{2}$

✔ Answer: $\frac{9a}{2}$

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#### 8) $\frac{5b}{3} - \frac{9b}{2}$

LCM of 3 and 2 is 6:
$$
= \frac{10b}{6} - \frac{27b}{6} = \frac{-17b}{6}
$$

✔ Answer: $-\frac{17b}{6}$

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#### 9) $\frac{1}{2y} + \frac{7}{y} = \frac{1}{2y} + \frac{14}{2y} = \frac{15}{2y}$

✔ Answer: $\frac{15}{2y}$

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#### 10) $\frac{6}{4k} - \frac{5}{6k}$

Simplify: $\frac{3}{2k} - \frac{5}{6k}$

LCM of 2 and 6 is 6:
$$
= \frac{9}{6k} - \frac{5}{6k} = \frac{4}{6k} = \frac{2}{3k}
$$

✔ Answer: $\frac{2}{3k}$

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#### 11) $\frac{r}{p} - \frac{4p}{q}$

Common denominator: $pq$
$$
= \frac{rq}{pq} - \frac{4p^2}{pq} = \frac{rq - 4p^2}{pq}
$$

✔ Answer: $\frac{rq - 4p^2}{pq}$

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#### 12) $\frac{x}{8} + \frac{1}{x + 2}$

Common denominator: $8(x + 2)$
$$
= \frac{x(x + 2)}{8(x + 2)} + \frac{8}{8(x + 2)} = \frac{x^2 + 2x + 8}{8(x + 2)}
$$

✔ Answer: $\frac{x^2 + 2x + 8}{8(x + 2)}$

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#### 13) $\frac{2}{x + 1} + \frac{1}{x + 2}$

Common denominator: $(x + 1)(x + 2)$
$$
= \frac{2(x + 2) + 1(x + 1)}{(x + 1)(x + 2)} = \frac{2x + 4 + x + 1}{(x + 1)(x + 2)} = \frac{3x + 5}{(x + 1)(x + 2)}
$$

✔ Answer: $\frac{3x + 5}{(x + 1)(x + 2)}$

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#### 14) $\frac{3}{y - 1} - \frac{1}{y + 2}$

Common denominator: $(y - 1)(y + 2)$
$$
= \frac{3(y + 2) - 1(y - 1)}{(y - 1)(y + 2)} = \frac{3y + 6 - y + 1}{(y - 1)(y + 2)} = \frac{2y + 7}{(y - 1)(y + 2)}
$$

✔ Answer: $\frac{2y + 7}{(y - 1)(y + 2)}$

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#### 15) $\frac{6}{1 - 2b} - \frac{b}{3 + b}$

Note: $1 - 2b = -(2b - 1)$, but let's just use LCD: $(1 - 2b)(3 + b)$

$$
= \frac{6(3 + b) - b(1 - 2b)}{(1 - 2b)(3 + b)} = \frac{18 + 6b - b + 2b^2}{(1 - 2b)(3 + b)} = \frac{2b^2 + 5b + 18}{(1 - 2b)(3 + b)}
$$

✔ Answer: $\frac{2b^2 + 5b + 18}{(1 - 2b)(3 + b)}$

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#### 16) $\frac{2}{x^2} - \frac{1}{x(x - 1)}$

Common denominator: $x^2(x - 1)$

$$
= \frac{2(x - 1)}{x^2(x - 1)} - \frac{x}{x^2(x - 1)} = \frac{2x - 2 - x}{x^2(x - 1)} = \frac{x - 2}{x^2(x - 1)}
$$

✔ Answer: $\frac{x - 2}{x^2(x - 1)}$

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#### 17) $7 - \frac{x - 4}{4x(x - 2)}$

Write $7 = \frac{7 \cdot 4x(x - 2)}{4x(x - 2)} = \frac{28x(x - 2)}{4x(x - 2)}$

But better: write $7 = \frac{7 \cdot 4x(x - 2)}{4x(x - 2)} = \frac{28x^2 - 56x}{4x(x - 2)}$

Then subtract:
$$
= \frac{28x^2 - 56x - (x - 4)}{4x(x - 2)} = \frac{28x^2 - 56x - x + 4}{4x(x - 2)} = \frac{28x^2 - 57x + 4}{4x(x - 2)}
$$

✔ Answer: $\frac{28x^2 - 57x + 4}{4x(x - 2)}$

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#### 18) $\frac{3y + 1}{(y - 2)^2} + \frac{y}{y - 2}$

Common denominator: $(y - 2)^2$

$$
= \frac{3y + 1 + y(y - 2)}{(y - 2)^2} = \frac{3y + 1 + y^2 - 2y}{(y - 2)^2} = \frac{y^2 + y + 1}{(y - 2)^2}
$$

✔ Answer: $\frac{y^2 + y + 1}{(y - 2)^2}$

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#### 19) $\frac{w}{w^2 - 1} + \frac{w + 1}{w - 1}$

Note: $w^2 - 1 = (w - 1)(w + 1)$

So:
$$
= \frac{w}{(w - 1)(w + 1)} + \frac{w + 1}{w - 1} = \frac{w + (w + 1)^2}{(w - 1)(w + 1)}
$$
Compute $(w + 1)^2 = w^2 + 2w + 1$

So:
$$
= \frac{w + w^2 + 2w + 1}{(w - 1)(w + 1)} = \frac{w^2 + 3w + 1}{(w - 1)(w + 1)}
$$

✔ Answer: $\frac{w^2 + 3w + 1}{(w - 1)(w + 1)}$

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#### 20) $\frac{1}{x^2 + 5x + 4} + \frac{1}{x^2 + 7x + 12}$

Factor:
- $x^2 + 5x + 4 = (x + 1)(x + 4)$
- $x^2 + 7x + 12 = (x + 3)(x + 4)$

Common denominator: $(x + 1)(x + 3)(x + 4)$

$$
= \frac{(x + 3) + (x + 1)}{(x + 1)(x + 3)(x + 4)} = \frac{2x + 4}{(x + 1)(x + 3)(x + 4)} = \frac{2(x + 2)}{(x + 1)(x + 3)(x + 4)}
$$

✔ Answer: $\frac{2(x + 2)}{(x + 1)(x + 3)(x + 4)}$

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#### 21) $\frac{2}{2x^2 + 5x - 3} - \frac{3}{3x^2 + 11x + 6}$

Factor both denominators:

- $2x^2 + 5x - 3$: Try factors of $2 \cdot (-3) = -6$: $6$ and $-1$
$2x^2 + 6x - x - 3 = 2x(x + 3) -1(x + 3) = (2x - 1)(x + 3)$

- $3x^2 + 11x + 6$: Factors of $3 \cdot 6 = 18$: $9$ and $2$
$3x^2 + 9x + 2x + 6 = 3x(x + 3) + 2(x + 3) = (3x + 2)(x + 3)$

So:
$$
\frac{2}{(2x - 1)(x + 3)} - \frac{3}{(3x + 2)(x + 3)}
$$

Common denominator: $(2x - 1)(3x + 2)(x + 3)$

$$
= \frac{2(3x + 2) - 3(2x - 1)}{(2x - 1)(3x + 2)(x + 3)}
$$

Numerator:
$$
2(3x + 2) = 6x + 4 \\
3(2x - 1) = 6x - 3 \\
\Rightarrow 6x + 4 - (6x - 3) = 6x + 4 - 6x + 3 = 7
$$

So:
$$
= \frac{7}{(2x - 1)(3x + 2)(x + 3)}
$$

✔ Answer: $\frac{7}{(2x - 1)(3x + 2)(x + 3)}$

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✔ Section B Summary:

| Problem | Answer |
|--------|--------|
| 1 | $\frac{x + 20}{5}$ |
| 2 | $\frac{y^2 - 6}{y}$ |
| 3 | $\frac{4x - 13}{x - 2}$ |
| 4 | $\frac{27w + 46}{3w + 5}$ |
| 5 | $\frac{x^2 + 8x + 15}{x + 2}$ |
| 6 | $\frac{8}{x}$ |
| 7 | $\frac{9a}{2}$ |
| 8 | $-\frac{17b}{6}$ |
| 9 | $\frac{15}{2y}$ |
| 10 | $\frac{2}{3k}$ |
| 11 | $\frac{rq - 4p^2}{pq}$ |
| 12 | $\frac{x^2 + 2x + 8}{8(x + 2)}$ |
| 13 | $\frac{3x + 5}{(x + 1)(x + 2)}$ |
| 14 | $\frac{2y + 7}{(y - 1)(y + 2)}$ |
| 15 | $\frac{2b^2 + 5b + 18}{(1 - 2b)(3 + b)}$ |
| 16 | $\frac{x - 2}{x^2(x - 1)}$ |
| 17 | $\frac{28x^2 - 57x + 4}{4x(x - 2)}$ |
| 18 | $\frac{y^2 + y + 1}{(y - 2)^2}$ |
| 19 | $\frac{w^2 + 3w + 1}{(w - 1)(w + 1)}$ |
| 20 | $\frac{2(x + 2)}{(x + 1)(x + 3)(x + 4)}$ |
| 21 | $\frac{7}{(2x - 1)(3x + 2)(x + 3)}$ |

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📌 Final Notes:

- Always factor when possible.
- Use common denominators carefully.
- Cancel common factors in numerators and denominators.
- Be cautious with signs, especially when subtracting.

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