The teaching of justification by faith alone is the teaching that all those who trust in God and His Messiah are declared righteous. You would not be declared righteous on the the basis of the merits of your good works. Although genuine faith will result in a change and in good works, although no one this side of heaven is perfected.
The Tanakh, the traditional Jewish Scriptures, fully supports this.
First, the Tanakh supports the problem. In Psalm 130 it is written, "If You, LORD, should mark iniquities, O Lord, who could stand?"
David, who knew a thing or two about sinning against God, wrote in Psalm 14:
The LORD looks down from heaven upon the children of men, to see if there are any who understand, who seek God. They have all turned aside, they have together become corrupt; there is none who does good, no, not one.
When Jeremiah describes the new covenant that was coming God says "I will forgive their iniquity, and their sin I will remmeber no more." (Jeremiah 31:34)
In the Torah, it is written: "Abram believed in the LORD, and He accounted it to him for righteousness." (Genesis 15:6)
The Torah does not read "Abram did a bunch of good stuff and it was accounted to him as righteousness." He was accounted righteous by faith. Later, Abraham's act of obedience proved his faith was genuine.
Speaking of the Messiah, Isaiah writes "by his knowledge my righteous servant will justify many, and he will bear their iniquities." (Isaiah 53:11)
We receive the Messiah's righteousness and we give him our iniquities. That is the exchange. Our filth for the Messiah's righteousness.
Zechariah shows this exchange:
Then he showed me Joshua the high priest standing before the angel of the LORD, and Satan standing at his right hand to accuse him. And the LORD said to Satan, "The LORD rebuke you, O Satan! The LORD who has chosen Jerusalem rebuke you! Is not this a brand plucked from the fire?" Now Joshua was standing before the angel, clothed with filthy garments. And the angel said to those who were standing before him, "Remove the filthy garments from him." And to him he said, "Behold, I have taken your iniquity away from you, and I will clothe you with pure vestments."
The Messiah takes away our iniquity and clothes us in his righteousness. That is truly good news.