To simplify the expression \(-\frac{3}{4}r + 3 - \frac{5}{6}r - 6\), let's first combine like terms.
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Combine the constant terms: \[ 3 - 6 = -3 \]
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Next, combine the \(r\) terms: \[ -\frac{3}{4}r - \frac{5}{6}r \] To do this, we need a common denominator. The least common multiple of 4 and 6 is 12. We can convert each fraction:
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For \(-\frac{3}{4}r\): \[ -\frac{3}{4}r = -\frac{3 \times 3}{4 \times 3}r = -\frac{9}{12}r \]
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For \(-\frac{5}{6}r\): \[ -\frac{5}{6}r = -\frac{5 \times 2}{6 \times 2}r = -\frac{10}{12}r \]
Now combine these: \[ -\frac{9}{12}r - \frac{10}{12}r = -\frac{19}{12}r \]
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Now, combine everything together: \[ -\frac{19}{12}r - 3 \]
Thus, the simplified expression is: \[ -\frac{19}{12}r - 3 \]