To solve the equation:

$\frac{1}{2}(x - 1) + 5 = \frac{1}{3}(2x + 1)$

Step 1: Eliminate the fractions

To eliminate the fractions, find the least common denominator (LCD) of 2 and 3, which is 6. Multiply every term by 6:

$6 \cdot \frac{1}{2}(x - 1) + 6 \cdot 5 = 6 \cdot \frac{1}{3}(2x + 1)$

Simplify each term:

$3(x - 1) + 30 = 2(2x + 1)$

Step 2: Expand and simplify

Expand both sides of the equation:

$3x - 3 + 30 = 4x + 2$

Simplify further:

$3x + 27 = 4x + 2$

Step 3: Solve for $x$

Isolate $x$ by subtracting $3x$ from both sides:

$27 = x + 2$

Then subtract 2 from both sides:

$x = 25$

Final Answer:

$x = 25$

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Would you like a detailed explanation of any step or have additional questions?

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Related Questions:

1. How do you solve equations with multiple fractions?
2. Can you explain how to check the solution of this equation?
3. What are some methods to simplify complex algebraic equations?
4. How does the LCD help in eliminating fractions?
5. What are common mistakes when solving equations with fractions?

Tip:

Always check your solution by substituting it back into the original equation to ensure it satisfies the equation.