To solve the compound inequality \( 9k + 7 \geq 18 + 10k \) or \( -17k - 8 + 15k \leq 10 \), we will solve each part separately.
Part 1: Solve \( 9k + 7 \geq 18 + 10k \)
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Start by subtracting \( 10k \) from both sides: \[ 9k - 10k + 7 \geq 18 \] Simplifying gives: \[ -k + 7 \geq 18 \]
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Next, subtract 7 from both sides: \[ -k \geq 11 \]
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Multiply by -1 (note that this reverses the inequality): \[ k \leq -11 \]
Part 2: Solve \( -17k - 8 + 15k \leq 10 \)
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Combine like terms on the left: \[ (-17k + 15k) - 8 \leq 10 \] This simplifies to: \[ -2k - 8 \leq 10 \]
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Add 8 to both sides: \[ -2k \leq 18 \]
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Divide by -2 (which reverses the inequality): \[ k \geq -9 \]
Combine results
Now we have two results:
- From the first part: \( k \leq -11 \)
- From the second part: \( k \geq -9 \)
Since the original inequality was an "or" situation, we combine these results:
The overall solution is: \[ k \leq -11 \quad \text{or} \quad k \geq -9 \]
The final answer in compound inequality form is: \[ (-\infty, -11] \cup [-9, \infty) \]
If you want the integer solutions only, they would be:
- \( k \leq -11 \) includes integers \( \ldots, -13, -12, -11 \)
- \( k \geq -9 \) includes integers \( -9, -8, -7, \ldots \)
Therefore, the integer solution set is: \[ k \in { \ldots, -13, -12, -11 } \cup { -9, -8, -7, \ldots } \]