Eli believed. But he did not believe enough to act. He and his family were cut off from the Lord.
Of course faith in Christ is a requirement--we cannot be saved without an atonement for our sins, nor can our hearts become pure without Him. But does faith alone bring salvation? Eli's story seems to indicate that faith without works is not enough. And yet, though Eli believed, he plainly didn't have very strong faith, or at least it wasn't strong enough to act.
Other scriptural examples such as Moses, Joshua, Peter and others seem to show that when faith is acted on, it gets stronger. Peter's faith in Christ seemed to grow with every challenge that he overcame. Trying then to separate faith from works is a difficult venture. Faith without works is weak and seems to dwindle. Faith with works grows and flourishes.
How strong must my faith be if I am to gain eternal life with my Heavenly Father and His Son? Christ commands me to become even as He is, and He has perfect faith. Eventually my faith must become that strong. Looks like I have a lot of work to do.