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Solve for w.

19–8w≥–5w+7>13–8w

1 answer

To solve the compound inequality \(19 - 8w \geq -5w + 7 > 13 - 8w\), we will break it into two parts:

  1. \(19 - 8w \geq -5w + 7\)
  2. \(-5w + 7 > 13 - 8w\)

Let's solve each part step by step.

Part 1: Solve \(19 - 8w \geq -5w + 7\)

  1. Rearrange the inequality: \[ 19 - 7 \geq -5w + 8w \] Simplifying the left side: \[ 12 \geq 3w \]

  2. Divide by 3: \[ 4 \geq w \quad \text{or} \quad w \leq 4 \]

Part 2: Solve \(-5w + 7 > 13 - 8w\)

  1. Rearrange the inequality: \[ -5w + 8w > 13 - 7 \] Simplifying gives: \[ 3w > 6 \]

  2. Divide by 3: \[ w > 2 \]

Combine the results

Now we have:

  1. \(w \leq 4\)
  2. \(w > 2\)

Combining these, we find that: \[ 2 < w \leq 4 \]

Final solution

The solution in interval notation is: \[ (2, 4] \]