Algebraic Fractions - Addition and Subtractions - Solving ... - Free Printable

Problem: Simplify the given algebraic fractions into a single fraction in its simplest form.

We will solve each problem step by step. Let's go through them one by one.

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#### 1. Simplify \( \frac{3}{5x} - \frac{1}{10x} \)

- Step 1: Find the least common denominator (LCD). The denominators are \(5x\) and \(10x\). The LCD is \(10x\).
- Step 2: Rewrite each fraction with the LCD:
\[
\frac{3}{5x} = \frac{3 \cdot 2}{5x \cdot 2} = \frac{6}{10x}
\]
\[
\frac{1}{10x} = \frac{1}{10x}
\]
- Step 3: Subtract the fractions:
\[
\frac{6}{10x} - \frac{1}{10x} = \frac{6 - 1}{10x} = \frac{5}{10x}
\]
- Step 4: Simplify the fraction:
\[
\frac{5}{10x} = \frac{1}{2x}
\]

Answer:
\[
\boxed{\frac{1}{2x}}
\]

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#### 2. Simplify \( \frac{8}{5x} - \frac{4}{15x} \)

- Step 1: Find the LCD. The denominators are \(5x\) and \(15x\). The LCD is \(15x\).
- Step 2: Rewrite each fraction with the LCD:
\[
\frac{8}{5x} = \frac{8 \cdot 3}{5x \cdot 3} = \frac{24}{15x}
\]
\[
\frac{4}{15x} = \frac{4}{15x}
\]
- Step 3: Subtract the fractions:
\[
\frac{24}{15x} - \frac{4}{15x} = \frac{24 - 4}{15x} = \frac{20}{15x}
\]
- Step 4: Simplify the fraction:
\[
\frac{20}{15x} = \frac{4}{3x}
\]

Answer:
\[
\boxed{\frac{4}{3x}}
\]

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#### 3. Simplify \( \frac{1}{4x} + \frac{1}{5x} \)

- Step 1: Find the LCD. The denominators are \(4x\) and \(5x\). The LCD is \(20x\).
- Step 2: Rewrite each fraction with the LCD:
\[
\frac{1}{4x} = \frac{1 \cdot 5}{4x \cdot 5} = \frac{5}{20x}
\]
\[
\frac{1}{5x} = \frac{1 \cdot 4}{5x \cdot 4} = \frac{4}{20x}
\]
- Step 3: Add the fractions:
\[
\frac{5}{20x} + \frac{4}{20x} = \frac{5 + 4}{20x} = \frac{9}{20x}
\]

Answer:
\[
\boxed{\frac{9}{20x}}
\]

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#### 4. Simplify \( \frac{2}{x} + \frac{3}{2x} \)

- Step 1: Find the LCD. The denominators are \(x\) and \(2x\). The LCD is \(2x\).
- Step 2: Rewrite each fraction with the LCD:
\[
\frac{2}{x} = \frac{2 \cdot 2}{x \cdot 2} = \frac{4}{2x}
\]
\[
\frac{3}{2x} = \frac{3}{2x}
\]
- Step 3: Add the fractions:
\[
\frac{4}{2x} + \frac{3}{2x} = \frac{4 + 3}{2x} = \frac{7}{2x}
\]

Answer:
\[
\boxed{\frac{7}{2x}}
\]

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#### 5. Simplify \( \frac{1}{x^2} + \frac{1}{x} \)

- Step 1: Find the LCD. The denominators are \(x^2\) and \(x\). The LCD is \(x^2\).
- Step 2: Rewrite each fraction with the LCD:
\[
\frac{1}{x^2} = \frac{1}{x^2}
\]
\[
\frac{1}{x} = \frac{1 \cdot x}{x \cdot x} = \frac{x}{x^2}
\]
- Step 3: Add the fractions:
\[
\frac{1}{x^2} + \frac{x}{x^2} = \frac{1 + x}{x^2}
\]

Answer:
\[
\boxed{\frac{1 + x}{x^2}}
\]

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#### 6. Simplify \( \frac{1}{xy} + \frac{1}{x} \)

- Step 1: Find the LCD. The denominators are \(xy\) and \(x\). The LCD is \(xy\).
- Step 2: Rewrite each fraction with the LCD:
\[
\frac{1}{xy} = \frac{1}{xy}
\]
\[
\frac{1}{x} = \frac{1 \cdot y}{x \cdot y} = \frac{y}{xy}
\]
- Step 3: Add the fractions:
\[
\frac{1}{xy} + \frac{y}{xy} = \frac{1 + y}{xy}
\]

Answer:
\[
\boxed{\frac{1 + y}{xy}}
\]

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#### 7. Simplify \( \frac{y}{x} - \frac{x}{y} \)

- Step 1: Find the LCD. The denominators are \(x\) and \(y\). The LCD is \(xy\).
- Step 2: Rewrite each fraction with the LCD:
\[
\frac{y}{x} = \frac{y \cdot y}{x \cdot y} = \frac{y^2}{xy}
\]
\[
\frac{x}{y} = \frac{x \cdot x}{y \cdot x} = \frac{x^2}{xy}
\]
- Step 3: Subtract the fractions:
\[
\frac{y^2}{xy} - \frac{x^2}{xy} = \frac{y^2 - x^2}{xy}
\]
- Step 4: Factor the numerator (difference of squares):
\[
y^2 - x^2 = (y - x)(y + x)
\]
So, the expression becomes:
\[
\frac{(y - x)(y + x)}{xy}
\]

Answer:
\[
\boxed{\frac{(y - x)(y + x)}{xy}}
\]

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#### 8. Simplify \( \frac{1}{xy} + \frac{1}{xz} + \frac{1}{yz} \)

- Step 1: Find the LCD. The denominators are \(xy\), \(xz\), and \(yz\). The LCD is \(xyz\).
- Step 2: Rewrite each fraction with the LCD:
\[
\frac{1}{xy} = \frac{1 \cdot z}{xy \cdot z} = \frac{z}{xyz}
\]
\[
\frac{1}{xz} = \frac{1 \cdot y}{xz \cdot y} = \frac{y}{xyz}
\]
\[
\frac{1}{yz} = \frac{1 \cdot x}{yz \cdot x} = \frac{x}{xyz}
\]
- Step 3: Add the fractions:
\[
\frac{z}{xyz} + \frac{y}{xyz} + \frac{x}{xyz} = \frac{z + y + x}{xyz}
\]

Answer:
\[
\boxed{\frac{x + y + z}{xyz}}
\]

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#### 9. Simplify \( \frac{1}{x} + \frac{1}{x+1} \)

- Step 1: Find the LCD. The denominators are \(x\) and \(x+1\). The LCD is \(x(x+1)\).
- Step 2: Rewrite each fraction with the LCD:
\[
\frac{1}{x} = \frac{1 \cdot (x+1)}{x \cdot (x+1)} = \frac{x+1}{x(x+1)}
\]
\[
\frac{1}{x+1} = \frac{1 \cdot x}{(x+1) \cdot x} = \frac{x}{x(x+1)}
\]
- Step 3: Add the fractions:
\[
\frac{x+1}{x(x+1)} + \frac{x}{x(x+1)} = \frac{(x+1) + x}{x(x+1)} = \frac{2x + 1}{x(x+1)}
\]

Answer:
\[
\boxed{\frac{2x + 1}{x(x+1)}}
\]

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#### 10. Simplify \( \frac{1}{x} - \frac{1}{x+1} \)

- Step 1: Find the LCD. The denominators are \(x\) and \(x+1\). The LCD is \(x(x+1)\).
- Step 2: Rewrite each fraction with the LCD:
\[
\frac{1}{x} = \frac{1 \cdot (x+1)}{x \cdot (x+1)} = \frac{x+1}{x(x+1)}
\]
\[
\frac{1}{x+1} = \frac{1 \cdot x}{(x+1) \cdot x} = \frac{x}{x(x+1)}
\]
- Step 3: Subtract the fractions:
\[
\frac{x+1}{x(x+1)} - \frac{x}{x(x+1)} = \frac{(x+1) - x}{x(x+1)} = \frac{1}{x(x+1)}
\]

Answer:
\[
\boxed{\frac{1}{x(x+1)}}
\]

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Final Answers for the First 10 Problems:

1. \(\boxed{\frac{1}{2x}}\)
2. \(\boxed{\frac{4}{3x}}\)
3. \(\boxed{\frac{9}{20x}}\)
4. \(\boxed{\frac{7}{2x}}\)
5. \(\boxed{\frac{1 + x}{x^2}}\)
6. \(\boxed{\frac{1 + y}{xy}}\)
7. \(\boxed{\frac{(y - x)(y + x)}{xy}}\)
8. \(\boxed{\frac{x + y + z}{xyz}}\)
9. \(\boxed{\frac{2x + 1}{x(x+1)}}\)
10. \(\boxed{\frac{1}{x(x+1)}}\)

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If you need solutions for the remaining problems, please let me know!