To simplify the expression \(-\frac{3}{4}r + 3 - \frac{5}{6}r - 6\), we will first combine like terms.
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Combine the constant terms: \(3 - 6\). \[ 3 - 6 = -3 \]
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Combine the \(r\) terms: \(-\frac{3}{4}r - \frac{5}{6}r\).
To combine the \(r\) terms, we need a common denominator. The least common multiple of \(4\) and \(6\) is \(12\).
- Convert \(-\frac{3}{4}r\) to have a denominator of \(12\): \[ -\frac{3}{4}r = -\frac{3 \times 3}{4 \times 3}r = -\frac{9}{12}r \]
- Convert \(-\frac{5}{6}r\) to have a denominator of \(12\): \[ -\frac{5}{6}r = -\frac{5 \times 2}{6 \times 2}r = -\frac{10}{12}r \]
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Now, combine these two converted terms: \[ -\frac{9}{12}r - \frac{10}{12}r = -\frac{9 + 10}{12}r = -\frac{19}{12}r \]
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Putting it all together, the simplified expression is: \[ -\frac{19}{12}r - 3 \]
Thus, the simplified expression is \(\boxed{-\frac{19}{12}r - 3}\).
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