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Solve for k.

9k+7≥18+10k or –17k–8+15k≤10
Write your answer as a compound inequality with integers.

1 answer

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 \( 9k + 7 \geq 18 + 10k \)

  1. Start by subtracting \( 10k \) from both sides: \[ 9k - 10k + 7 \geq 18 \] Simplifying gives: \[ -k + 7 \geq 18 \]

  2. Next, subtract 7 from both sides: \[ -k \geq 11 \]

  3. Multiply by -1 (note that this reverses the inequality): \[ k \leq -11 \]

Part 2: Solve \( -17k - 8 + 15k \leq 10 \)

  1. Combine like terms on the left: \[ (-17k + 15k) - 8 \leq 10 \] This simplifies to: \[ -2k - 8 \leq 10 \]

  2. Add 8 to both sides: \[ -2k \leq 18 \]

  3. 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: \[ k \in { \ldots, -13, -12, -11 } \cup { -9, -8, -7, \ldots } \]