To solve the equation \( \frac{1}{2}x + \frac{3}{2}(x + 1) - \frac{1}{4} = 5 \), we will follow these steps:
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First, distribute \( \frac{3}{2} \) to the terms in the parentheses: \[ \frac{3}{2}(x + 1) = \frac{3}{2}x + \frac{3}{2} \] So the equation becomes: \[ \frac{1}{2}x + \frac{3}{2}x + \frac{3}{2} - \frac{1}{4} = 5 \]
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Now combine like terms on the left side: \[ \frac{1}{2}x + \frac{3}{2}x = \frac{4}{2}x = 2x \] Hence, we can rewrite the equation as: \[ 2x + \frac{3}{2} - \frac{1}{4} = 5 \]
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Next, simplify \( \frac{3}{2} - \frac{1}{4} \). To do this, we can convert \( \frac{3}{2} \) to have a denominator of 4: \[ \frac{3}{2} = \frac{6}{4} \] Now we can subtract: \[ \frac{6}{4} - \frac{1}{4} = \frac{5}{4} \] So our equation now is: \[ 2x + \frac{5}{4} = 5 \]
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Next, isolate \( 2x \) by subtracting \( \frac{5}{4} \) from both sides: \[ 2x = 5 - \frac{5}{4} \] We convert 5 into quarters: \[ 5 = \frac{20}{4} \] Thus, \[ 2x = \frac{20}{4} - \frac{5}{4} = \frac{15}{4} \]
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Finally, solve for \( x \) by dividing both sides by 2: \[ x = \frac{15}{4} \cdot \frac{1}{2} = \frac{15}{8} \]
The solution for \( x \) is: \[ \boxed{\frac{15}{8}} \]