To solve the equation \( \frac{17}{3} \left( x - \frac{3}{2} \right) = -\frac{5}{4} \) using the distributive property, we first distribute \( \frac{17}{3} \) to both terms inside the parentheses.
\[ \frac{17}{3} x - \frac{17}{3} \cdot \frac{3}{2} = -\frac{5}{4} \]
Calculating \( \frac{17}{3} \cdot \frac{3}{2} \):
\[ \frac{17 \cdot 3}{3 \cdot 2} = \frac{17}{2} \]
Substituting this back into the equation, we have:
\[ \frac{17}{3} x - \frac{17}{2} = -\frac{5}{4} \]
Next, we want to isolate \( \frac{17}{3} x \). To do this, we can add \( \frac{17}{2} \) to both sides:
\[ \frac{17}{3} x = -\frac{5}{4} + \frac{17}{2} \]
To add these fractions, we need a common denominator. The least common multiple of \( 4 \) and \( 2 \) is \( 4 \). We can convert \( \frac{17}{2} \) to have this common denominator:
\[ \frac{17}{2} = \frac{17 \cdot 2}{2 \cdot 2} = \frac{34}{4} \]
Now, substituting this back in, we have:
\[ \frac{17}{3} x = -\frac{5}{4} + \frac{34}{4} \]
Now we can combine the fractions on the right:
\[ \frac{17}{3} x = \frac{34 - 5}{4} = \frac{29}{4} \]
Next, we can solve for \( x \) by multiplying both sides of the equation by the reciprocal of \( \frac{17}{3} \), which is \( \frac{3}{17} \):
\[ x = \frac{29}{4} \cdot \frac{3}{17} \]
Calculating this gives:
\[ x = \frac{29 \cdot 3}{4 \cdot 17} = \frac{87}{68} \]
Thus, the solution to the equation \( \frac{17}{3} \left( x - \frac{3}{2} \right) = -\frac{5}{4} \) is:
\[ \boxed{\frac{87}{68}} \]
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