This question tests recognition of which antibiotic classes actually reach gram-negative targets like Pseudomonas.
- Ceftazidime: a cephalosporin built for anti-pseudomonal cover, so it is effective and not the answer.
- Piperacillin: an extended-spectrum penicillin with established Pseudomonas activity, so it is effective too.
- Vancomycin: a glycopeptide restricted to gram-positive organisms because it cannot cross the gram-negative outer membrane — it has no activity against Pseudomonas, making it the correct answer.
- Aztreonam: a monobactam active specifically against gram-negative aerobes including Pseudomonas, so it is effective and not the answer.
Correct answer: Vancomycin, since it works only on gram-positive bacteria and cannot penetrate Pseudomonas.