To simplify the expression \(-\frac{3}{4} r + 3 - \frac{5}{6} r - 6\), follow these steps:
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Combine the constant terms \(3\) and \(-6\): \[ 3 - 6 = -3 \]
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Combine the \(r\) terms \(-\frac{3}{4} r\) and \(-\frac{5}{6} r\): To combine these fractions, find a common denominator. The least common multiple of \(4\) and \(6\) is \(12\).
Convert \(-\frac{3}{4} r\) to an equivalent fraction with 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 an equivalent fraction with 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 the fractions: \[ -\frac{9}{12} r - \frac{10}{12} r = -\frac{19}{12} r \]
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Putting it all together, we have: \[ -\frac{19}{12} r - 3 \]
Thus, the simplified expression is: \[ -\frac{19}{12} r - 3 \]