"To eradicate all suffering" is a reiteration of "to attain deliverance from all suffering" as stated in the beginning of the Sutra. It is the great vow of Avalokitesvara Bodhisattva.
All sufferings including the Three Sufferings and Eight Sufferings, can be eradicated by the Prajna-paramita mantra. If one practices the Prajna-paramita, one can cross beyond the Three Worlds and depart from the sufferings of incarnation of birth and death.
Buddha is a speaker of true words, real words, non-exaggerated words and consistent words, therefore the Heart Sutra that Shakyamuni preached is not false. Everybody should have faith on it, interpret and think over it, and practice diligently, thus verify whether it is true or false.