To solve the expression \(\frac{\frac{8}{9}x - \frac{3}{4}}{\frac{1}{9}x - \frac{4}{5} \div \frac{3}{5}}\), we will first simplify the expression in the denominator.
Step 1: Simplify the denominator
The denominator contains a division:
\[ \frac{4}{5} \div \frac{3}{5} \]
Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \frac{4}{5} \div \frac{3}{5} = \frac{4}{5} \times \frac{5}{3} = \frac{4}{3} \]
Step 2: Rewrite the entire expression
Now we can rewrite the entire expression with the simplified denominator:
\[ \frac{\frac{8}{9}x - \frac{3}{4}}{\frac{1}{9}x - \frac{4}{3}} \]
Step 3: Get rid of fractions
To eliminate fractions, we can multiply both the numerator and the denominator by the least common multiple (LCM) of the denominators. The denominators in the numerator and denominator of our expression are \(9\), \(4\), and \(3\). The LCM of these numbers is \(36\).
Multiply the numerator and the denominator by \(36\):
\[ \frac{36\left(\frac{8}{9}x - \frac{3}{4}\right)}{36\left(\frac{1}{9}x - \frac{4}{3}\right)} \]
Step 4: Distribute \(36\)
Now, we can simplify each part:
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For the numerator: \[ 36 \left(\frac{8}{9}x - \frac{3}{4}\right) = 36 \cdot \frac{8}{9}x - 36 \cdot \frac{3}{4} = 4 \cdot 8x - 27 = 32x - 27 \]
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For the denominator: \[ 36 \left(\frac{1}{9}x - \frac{4}{3}\right) = 36 \cdot \frac{1}{9}x - 36 \cdot \frac{4}{3} = 4x - 48 \]
Step 5: Write the final simplified expression
Now, the expression is simplified to:
\[ \frac{32x - 27}{4x - 48} \]
This is the final simplified form of the given expression.