Solving equations involving algebraic fractions with detailed solutions.

Problem: Solve the following equations involving algebraic fractions.

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

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#### a) \( \frac{x}{3} - \frac{x}{4} = 2 \)

1. Find a common denominator: The denominators are 3 and 4. The least common denominator (LCD) is 12.
2. Rewrite the fractions with the common denominator:
\[
\frac{x}{3} = \frac{4x}{12}, \quad \frac{x}{4} = \frac{3x}{12}
\]
So the equation becomes:
\[
\frac{4x}{12} - \frac{3x}{12} = 2
\]
3. Combine the fractions:
\[
\frac{4x - 3x}{12} = 2 \implies \frac{x}{12} = 2
\]
4. Solve for \( x \):
\[
x = 2 \times 12 \implies x = 24
\]

Answer: \( x = 24 \)

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#### b) \( \frac{y}{2} + \frac{y}{3} = -5 \)

1. Find a common denominator: The denominators are 2 and 3. The least common denominator (LCD) is 6.
2. Rewrite the fractions with the common denominator:
\[
\frac{y}{2} = \frac{3y}{6}, \quad \frac{y}{3} = \frac{2y}{6}
\]
So the equation becomes:
\[
\frac{3y}{6} + \frac{2y}{6} = -5
\]
3. Combine the fractions:
\[
\frac{3y + 2y}{6} = -5 \implies \frac{5y}{6} = -5
\]
4. Solve for \( y \):
\[
5y = -5 \times 6 \implies 5y = -30 \implies y = \frac{-30}{5} \implies y = -6
\]

Answer: \( y = -6 \)

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#### c) \( \frac{a}{6} - \frac{2a}{3} = 3 \)

1. Find a common denominator: The denominators are 6 and 3. The least common denominator (LCD) is 6.
2. Rewrite the fractions with the common denominator:
\[
\frac{a}{6} = \frac{a}{6}, \quad \frac{2a}{3} = \frac{4a}{6}
\]
So the equation becomes:
\[
\frac{a}{6} - \frac{4a}{6} = 3
\]
3. Combine the fractions:
\[
\frac{a - 4a}{6} = 3 \implies \frac{-3a}{6} = 3
\]
4. Simplify the fraction:
\[
\frac{-3a}{6} = \frac{-a}{2} \implies \frac{-a}{2} = 3
\]
5. Solve for \( a \):
\[
-a = 3 \times 2 \implies -a = 6 \implies a = -6
\]

Answer: \( a = -6 \)

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

1. Find a common denominator: The denominators are 2 and 3. The least common denominator (LCD) is 6.
2. Rewrite the fractions with the common denominator:
\[
\frac{x+1}{2} = \frac{3(x+1)}{6}, \quad \frac{x-1}{3} = \frac{2(x-1)}{6}
\]
So the equation becomes:
\[
\frac{3(x+1)}{6} + \frac{2(x-1)}{6} = 1
\]
3. Combine the fractions:
\[
\frac{3(x+1) + 2(x-1)}{6} = 1
\]
4. Simplify the numerator:
\[
3(x+1) + 2(x-1) = 3x + 3 + 2x - 2 = 5x + 1
\]
So the equation becomes:
\[
\frac{5x + 1}{6} = 1
\]
5. Solve for \( x \):
\[
5x + 1 = 1 \times 6 \implies 5x + 1 = 6 \implies 5x = 6 - 1 \implies 5x = 5 \implies x = 1
\]

Answer: \( x = 1 \)

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#### e) \( \frac{5}{6x} - \frac{1}{x+1} = \frac{1}{3x} \)

1. Find a common denominator: The denominators are \( 6x \), \( x+1 \), and \( 3x \). The least common denominator (LCD) is \( 6x(x+1) \).
2. Rewrite the fractions with the common denominator:
\[
\frac{5}{6x} = \frac{5(x+1)}{6x(x+1)}, \quad \frac{1}{x+1} = \frac{6x}{6x(x+1)}, \quad \frac{1}{3x} = \frac{2(x+1)}{6x(x+1)}
\]
So the equation becomes:
\[
\frac{5(x+1)}{6x(x+1)} - \frac{6x}{6x(x+1)} = \frac{2(x+1)}{6x(x+1)}
\]
3. Combine the fractions:
\[
\frac{5(x+1) - 6x}{6x(x+1)} = \frac{2(x+1)}{6x(x+1)}
\]
4. Simplify the numerators:
\[
5(x+1) - 6x = 5x + 5 - 6x = -x + 5
\]
So the equation becomes:
\[
\frac{-x + 5}{6x(x+1)} = \frac{2(x+1)}{6x(x+1)}
\]
5. Since the denominators are the same, equate the numerators:
\[
-x + 5 = 2(x+1)
\]
6. Solve for \( x \):
\[
-x + 5 = 2x + 2 \implies 5 - 2 = 2x + x \implies 3 = 3x \implies x = 1
\]

Answer: \( x = 1 \)

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Final Answers:

\[
\boxed{24, -6, -6, 1, 1}
\]