To solve the compound inequalities, we will break them down into two separate inequalities:
- \( \frac{n - 19}{3} \geq -3 \)
- \( \frac{5n + 18}{3} \leq 6 \)
Solving the first inequality:
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Start with: \[ \frac{n - 19}{3} \geq -3 \]
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Multiply both sides by 3 (since 3 is positive, the inequality direction stays the same): \[ n - 19 \geq -9 \]
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Add 19 to both sides: \[ n \geq 10 \]
Solving the second inequality:
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Start with: \[ \frac{5n + 18}{3} \leq 6 \]
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Multiply both sides by 3: \[ 5n + 18 \leq 18 \]
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Subtract 18 from both sides: \[ 5n \leq 0 \]
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Divide both sides by 5 (again, 5 is positive, so the inequality direction stays the same): \[ n \leq 0 \]
Final Expression:
Now, we combine the results of both inequalities to form a compound inequality:
\[ n \geq 10 \quad \text{or} \quad n \leq 0 \]
Thus, the solution is expressed as: \[ n \in (-\infty, 0] \cup [10, \infty) \]
In terms of integers, the solution can be written as:
\[ n \leq 0 \quad \text{or} \quad n \geq 10 \]
This describes the set of all integers that satisfy either of the conditions.