To solve the inequality \( 34 \frac{1}{2} < 8x + 25 \frac{1}{2} \), we first convert the mixed numbers to improper fractions or decimal form for easier calculations.
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Convert \( 34 \frac{1}{2} \): \[ 34 \frac{1}{2} = 34 + \frac{1}{2} = 34.5 \]
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Convert \( 25 \frac{1}{2} \): \[ 25 \frac{1}{2} = 25 + \frac{1}{2} = 25.5 \]
Now, substitute these values into the inequality: \[ 34.5 < 8x + 25.5 \]
Next, isolate \( 8x \) by subtracting \( 25.5 \) from both sides: \[ 34.5 - 25.5 < 8x \]
This simplifies to: \[ 9 < 8x \]
Now divide both sides by \( 8 \): \[ \frac{9}{8} < x \]
This can also be written as: \[ x > \frac{9}{8} \]
To express it as a decimal, \( \frac{9}{8} = 1.125 \).
Therefore, the solution to the inequality is: \[ x > \frac{9}{8} \quad \text{or} \quad x > 1.125 \]