To find the least common denominator (LCD) for the fractions in the equation \(\frac{3}{4}(x - 3) - \frac{1}{2} = \frac{2}{3}\), we first identify the denominators: 4, 2, and 3.
Next, we will find the least common multiple (LCM) of these denominators:
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The prime factorization of each of the denominators:
- \(4 = 2^2\)
- \(2 = 2^1\)
- \(3 = 3^1\)
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For the LCM, we take the highest power of each prime number that appears in the factorizations:
- The highest power of \(2\) is \(2^2\),
- The highest power of \(3\) is \(3^1\).
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Now, we multiply these together: \[ \text{LCM} = 2^2 \times 3^1 = 4 \times 3 = 12 \]
Thus, the least common denominator of the equation is \(\boxed{12}\).
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