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: \(3 - 6\). \[ 3 - 6 = -3 \]
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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 \(12\).
Convert \(-\frac{3}{4}r\) to a fraction with a denominator of \(12\): \[ -\frac{3}{4} = -\frac{3 \times 3}{4 \times 3} = -\frac{9}{12} \]
Convert \(-\frac{5}{6}r\) to a fraction with a denominator of \(12\): \[ -\frac{5}{6} = -\frac{5 \times 2}{6 \times 2} = -\frac{10}{12} \]
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Now, combine the \(r\) terms: \[ -\frac{9}{12}r - \frac{10}{12}r = -\left(\frac{9 + 10}{12}\right)r = -\frac{19}{12}r \]
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Now place together the constant term and the combined \(r\) term: \[ -\frac{19}{12}r - 3 \]
So, the simplified expression is: \[ -\frac{19}{12}r - 3 \]
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