To divide the fractions \( \frac{4 - 6x}{24} \) and \( \frac{2 - 3x}{8} \), we can multiply the first fraction by the reciprocal of the second fraction.
The problem can be rewritten as follows:
\[ \frac{4 - 6x}{24} \div \frac{2 - 3x}{8} = \frac{4 - 6x}{24} \times \frac{8}{2 - 3x} \]
Now, let's simplify this step by step.
- Multiply the numerators and the denominators:
\[ = \frac{(4 - 6x) \times 8}{24 \times (2 - 3x)} \]
- Simplify the expression:
- First, simplify the numerator:
\[ (4 - 6x) \times 8 = 32 - 48x \]
- Now, the denominator:
\[ 24 \times (2 - 3x) = 48 - 72x \]
So, we have:
\[ \frac{32 - 48x}{48 - 72x} \]
- Now, let's simplify further. Factor the numerator and denominator:
The numerator \( 32 - 48x \) can be factored:
\[ = 16(2 - 3x) \]
The denominator \( 48 - 72x \) can also be factored:
\[ = 24(2 - 3x) \]
- Now substituting back:
\[ \frac{16(2 - 3x)}{24(2 - 3x)} \]
- Cancel common factors:
Since \( 2 - 3x \) is common in both the numerator and the denominator (assuming \( 2 - 3x \neq 0 \)), we can cancel it out:
\[ = \frac{16}{24} \]
- Finally, simplify the fraction \( \frac{16}{24} \):
\[ = \frac{2}{3} \]
Thus, the simplified answer is:
\[ \frac{2}{3} \]
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