To solve the inequality \( 34 \frac{1}{2} > 8x + 25 \frac{1}{2} \), we first convert the mixed numbers into improper fractions or decimals for easier manipulation.
-
Convert \( 34 \frac{1}{2} \): \[ 34 \frac{1}{2} = 34 + \frac{1}{2} = \frac{68}{2} + \frac{1}{2} = \frac{69}{2} \]
-
Convert \( 25 \frac{1}{2} \): \[ 25 \frac{1}{2} = 25 + \frac{1}{2} = \frac{50}{2} + \frac{1}{2} = \frac{51}{2} \]
Now, let's rewrite the inequality: \[ \frac{69}{2} > 8x + \frac{51}{2} \]
Next, we can eliminate the fractions by multiplying the entire inequality by 2: \[ 69 > 16x + 51 \]
Now we will isolate \( x \):
-
Subtract 51 from both sides: \[ 69 - 51 > 16x \] \[ 18 > 16x \]
-
Divide both sides by 16: \[ \frac{18}{16} > x \] \[ \frac{9}{8} > x \] or equivalently, \[ x < \frac{9}{8} \]
Thus, the solution to the inequality is: \[ x < \frac{9}{8} \]