Learn Portuguese (Portugal): Ele precisa de elevar a cadeira, de modo a aliviar o pulso. - He needs to elevate the chair, in order to relieve the wrist.

While para aliviar is perfectly correct and very common, de modo a (or de forma a) is slightly more formal. It emphasizes the manner or method of achieving the goal, translating closely to 'so as to' or 'in such a way as to'. Both are followed by the infinitive verb.

If the goal was to relieve someone else's wrist, would we still use de modo a?

No. We use de modo a plus the infinitive (aliviar) because the subject doing the elevating is the same as the subject feeling the relief. If he raised the chair for someone else, the subjects would be different, requiring de modo que plus the subjunctive mood (for example, de modo que ela alivie).

Why doesn't the sentence say o seu pulso to mean 'his wrist'?

In Portuguese, it is standard to use the definite article (o or a) instead of a possessive pronoun for body parts, clothing, and personal items when the owner is obvious from the context. Since he is the subject of the sentence, o pulso naturally means his own wrist.

Could I use levantar instead of elevar here?