To solve the problem \(-2 \frac{1}{2} + 6 \frac{2}{3}\), we follow these steps:
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Convert the mixed numbers into improper fractions.
For \(-2 \frac{1}{2}\): \[ -2 \frac{1}{2} = -\left(2 \times 2 + 1\right) / 2 = -\frac{5}{2} \]
For \(6 \frac{2}{3}\): \[ 6 \frac{2}{3} = 6 \times 3 + 2 / 3 = \frac{18 + 2}{3} = \frac{20}{3} \]
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Now we have: \[ -\frac{5}{2} + \frac{20}{3} \]
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Before we can add these fractions, we need a common denominator. The denominators are 2 and 3. The least common denominator (LCD) of 2 and 3 is 6.
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Convert both fractions to have the common denominator of 6.
For \(-\frac{5}{2}\): \[ -\frac{5}{2} = -\frac{5 \times 3}{2 \times 3} = -\frac{15}{6} \]
For \(\frac{20}{3}\): \[ \frac{20}{3} = \frac{20 \times 2}{3 \times 2} = \frac{40}{6} \]
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Now we can add the fractions: \[ -\frac{15}{6} + \frac{40}{6} = \frac{-15 + 40}{6} = \frac{25}{6} \]
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The result can be expressed as a mixed number: \[ \frac{25}{6} = 4 \frac{1}{6} \]
Thus, the final answer is: \[ \boxed{4 \frac{1}{6}} \]