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Solve for b.

7+14b≥b+7>14b–6
Write your answer as a compound inequality with integers.

1 answer

To solve the compound inequality \( 7 + 14b \geq b + 7 > 14b - 6 \), we will break it into two parts:

  1. \( 7 + 14b \geq b + 7 \)
  2. \( b + 7 > 14b - 6 \)

Step 1: Solve the first inequality \( 7 + 14b \geq b + 7 \)

  1. Subtract \( b \) and \( 7 \) from both sides: \[ 7 + 14b - b - 7 \geq 0 \] \[ 13b \geq 0 \]

  2. Divide both sides by \( 13 \): \[ b \geq 0 \]

Step 2: Solve the second inequality \( b + 7 > 14b - 6 \)

  1. Subtract \( b \) from both sides: \[ 7 > 13b - 6 \]

  2. Add \( 6 \) to both sides: \[ 13 > 13b \]

  3. Divide both sides by \( 13 \): \[ 1 > b \quad \text{or} \quad b < 1 \]

Step 3: Combine the results

From the two steps, we have:

  1. \( b \geq 0 \)
  2. \( b < 1 \)

Final Result

Combining these, we get the compound inequality: \[ 0 \leq b < 1 \]

In interval notation, this is represented as: \[ [0, 1) \]

So the solution for \( b \) is: \[ \boxed{[0, 1)} \]