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This piece of text focuses on the written evidence that we as human beings are obligated to include all of God's creations, specifically those in nature, in our care. To guard and maintain its health and beauty.

(טו) וַיִּקַּ֛ח יְהוָ֥ה אֱלֹהִ֖ים אֶת־הָֽאָדָ֑ם וַיַּנִּחֵ֣הוּ בְגַן־עֵ֔דֶן לְעָבְדָ֖הּ וּלְשָׁמְרָֽהּ׃
(15) The LORD God took the man and placed him in the garden of Eden, to till it and tend it.
ויקח ה' אלהים, אע"פ שזכר למעלה וישם שם, חזר ושנה עוד ואמר ויקח, לפי שרוצה לזכור המצוה שצוהו האל.

ויקח ה' אלוקים, even though we have been told already in verse 8 that G’d placed Adam in Gan Eden, the Torah repeats this by stating that “G’d took man etc.,” because at this point the Torah wanted to inform us of the commandment which accompanied Adam’s taking up residence in that idyllic place. Our verse also spells out something that had not been mentioned previously, i.e. that Adam was not to treat Gan Eden as a retreat, but that he was made responsible for maintaining it in good order.

Living a life through tikkun olam is being concerned for the welfare of others, and to not hold any forms of hatred in your heart, for everyone is in God's image.

(טז) לֹא־תֵלֵ֤ךְ רָכִיל֙ בְּעַמֶּ֔יךָ לֹ֥א תַעֲמֹ֖ד עַל־דַּ֣ם רֵעֶ֑ךָ אֲנִ֖י יקוק׃ (יז) לֹֽא־תִשְׂנָ֥א אֶת־אָחִ֖יךָ בִּלְבָבֶ֑ךָ הוֹכֵ֤חַ תּוֹכִ֙יחַ֙ אֶת־עֲמִיתֶ֔ךָ וְלֹא־תִשָּׂ֥א עָלָ֖יו חֵֽטְא׃
(16) Do not deal basely with your countrymen. Do not profit by the blood of your fellow: I am the LORD. (17) You shall not hate your kinsfolk in your heart. Reprove your kinsman but incur no guilt because of him.

Justice and righteousness are connected to each other in Jewish tradition. When we study, speak, and act towards justice, we continue God's work and keep repairing the world.

(כ) צֶ֥דֶק צֶ֖דֶק תִּרְדֹּ֑ף לְמַ֤עַן תִּֽחְיֶה֙ וְיָרַשְׁתָּ֣ אֶת־הָאָ֔רֶץ אֲשֶׁר־יְהוָ֥ה אֱלֹהֶ֖יךָ נֹתֵ֥ן לָֽךְ׃ (ס)
(20) Justice, justice shall you pursue, that you may thrive and occupy the land that the LORD your God is giving you.
צדק צדק תרדוף. ע"ד הפשט יזהיר שיצדק האדם במעשיו ובדבריו, כי אלה הם הדברים שאפשר לו לאדם להרע לעצמו ולאחרים, ועל כן בא כפל הלשון, וכל הצודק בדבריו אות ומופת על צדקת מעשיו, ועל כן ראוי לכל אחד ואחד מישראל שיצדק בדבור ובמעשה, כענין שכתוב (צפניה ג׳:י״ג) שארית ישראל לא יעשו עולה ולא ידברו כזב. או ידבר הכתוב עם האנשים אשר להם הריב ויזהיר אותן לרדוף אחרי הצדק הן לריוח הן להפסד, וזהו כפל הלשון.
צדק צדק תרדוף, “Righteousness, righteousness, you shall pursue.” According to the plain meaning of the text the Torah warns (by repeating) that one must strive to be righteous both in word and in deed. These are the two ways in which one may potentially inflict harm upon both oneself and upon others. Everyone who speaks righteously reflects the fact that his deeds are most likely righteous also; this is why it behooves every Jew to be both righteous in his speech and in his deeds. This sentiment is reflected in Tzefaniah 3,13 when he said of the remnant of the people of Israel that “they shall do no wrong or speak falsehood; a deceitful tongue shall not be in their mouths.”
Alternatively, our verse addresses the people who are subject to litigation and exhorts them to strive for righteousness regardless of whether this will be financially beneficial or harmful to them. This is why the Torah repeats the exhortation.
(יז) לִמְד֥וּ הֵיטֵ֛ב דִּרְשׁ֥וּ מִשְׁפָּ֖ט אַשְּׁר֣וּ חָמ֑וֹץ שִׁפְט֣וּ יָת֔וֹם רִ֖יבוּ אַלְמָנָֽה׃ (ס)
(17) Learn to do good. Devote yourselves to justice; Aid the wronged. Uphold the rights of the orphan; Defend the cause of the widow.
למדו. רפי נקוד והוא מגזרת למוד היו למדין להטיב המלמד לעצמו הוא ממשקולת ל' קל לפיכך צווי ל' רבים שלו נקוד בחירק כמו אמרו שמעו אבל המלמ' לאחרי' הוא ממשקולת ל' כבד המדגיש ואם בא לצוות את הרבים הוא נקוד למדו וכן דרשו מגזרת דרוש אבל אשרו שהשי"ן מודגש' ממשקולת ל' כבד ומגזרת אשר לפיכך ל' צווי לרבים נקוד פתח (כמו אשרו דברו ספרו בשרו):
Learn It is punctuated ‘raphe,’ weak, without a dagesh. This is from the form לָמֹד, learn to do good. One who teaches himself is of the ‘kal’ form. Therefore, its imperative plural is voweled with a ‘chirik’ like אִמְרוּ, שִׁמְעוּ, but one who teaches others is of the form of the ‘heavy conjugation’ (pi’el) with a ‘dagesh,’ and if one comes to command a number of people, the word is voweled לַמְּדוּ. And so, דִּרְשׁוּ, from the form דְרשׁ, but אַשְּׁרוּ in which the ‘shin’ has a ‘dagesh,’ is from the ‘heavy conjugation,’ and from the form אַשֵּׁר ; therefore, the imperative plural is voweled with a ‘patach’ like בַּשְּׂרוּ, סַפְּרוּ, דַּבְּרוּ.

וכבר היה רבי טרפון וזקנים מסובין בעלית בית נתזה בלוד נשאלה שאילה זו בפניהם תלמוד גדול או מעשה גדול נענה רבי טרפון ואמר מעשה גדול נענה ר"ע ואמר תלמוד גדול נענו כולם ואמרו תלמוד גדול שהתלמוד מביא לידי מעשה.

Kiddushin 40b:8
Rabbi Tarfon and some elders were reclining in an upper chamber in the house of Nitza in Lod when this question came up: Which is greater, study or action? Rabbi Tarfon spoke up and said: Action is greater. Rabbi Akiva spoke up and said: Study is greater. The others then spoke up and said: Study is greater because it leads to action. [Soncino translation]

When we commit harm, as interpreted by the Mishnah in connection to Abel's murder by Cain, the injustices we commit cause further harm to the world. It is a reminder that tikkun olam does the opposite and that each good deed helps fix our broken world.

וַיֹּ֖אמֶר מֶ֣ה עָשִׂ֑יתָ ק֚וֹל דְּמֵ֣י אָחִ֔יךָ צֹעֲקִ֥ים אֵלַ֖י מִן־הָֽאֲדָמָֽה׃
Then He said, “What have you done? Hark, your brother’s blood cries out to Me from the ground!
כֵּיצַד מְאַיְּמִין אֶת הָעֵדִים עַל עֵדֵי נְפָשׁוֹת, הָיוּ מַכְנִיסִין אוֹתָן וּמְאַיְּמִין עֲלֵיהֶן. שֶׁמָּא תֹאמְרוּ מֵאֹמֶד, וּמִשְּׁמוּעָה, עֵד מִפִּי עֵד וּמִפִּי אָדָם נֶאֱמָן שָׁמַעְנוּ, אוֹ שֶׁמָּא אִי אַתֶּם יוֹדְעִין שֶׁסּוֹפֵנוּ לִבְדֹּק אֶתְכֶם בִּדְרִישָׁה וּבַחֲקִירָה. הֱווּ יוֹדְעִין שֶׁלֹּא כְדִינֵי מָמוֹנוֹת דִּינֵי נְפָשׁוֹת. דִּינֵי מָמוֹנוֹת, אָדָם נוֹתֵן מָמוֹן וּמִתְכַּפֵּר לוֹ. דִּינֵי נְפָשׁוֹת, דָּמוֹ וְדַם זַרְעִיּוֹתָיו תְּלוּיִין בּוֹ עַד סוֹף הָעוֹלָם, שֶׁכֵּן מָצִינוּ בְקַיִן שֶׁהָרַג אֶת אָחִיו, שֶׁנֶּאֱמַר (בראשית ד) דְּמֵי אָחִיךָ צֹעֲקִים, אֵינוֹ אוֹמֵר דַּם אָחִיךָ אֶלָּא דְּמֵי אָחִיךָ, דָּמוֹ וְדַם זַרְעִיּוֹתָיו. דָּבָר אַחֵר, דְּמֵי אָחִיךָ, שֶׁהָיָה דָמוֹ מֻשְׁלָךְ עַל הָעֵצִים וְעַל הָאֲבָנִים. לְפִיכָךְ נִבְרָא אָדָם יְחִידִי, לְלַמֶּדְךָ, שֶׁכָּל הַמְאַבֵּד נֶפֶשׁ אַחַת מִיִּשְׂרָאֵל, מַעֲלֶה עָלָיו הַכָּתוּב כְּאִלּוּ אִבֵּד עוֹלָם מָלֵא. וְכָל הַמְקַיֵּם נֶפֶשׁ אַחַת מִיִּשְׂרָאֵל, מַעֲלֶה עָלָיו הַכָּתוּב כְּאִלּוּ קִיֵּם עוֹלָם מָלֵא. וּמִפְּנֵי שְׁלוֹם הַבְּרִיּוֹת, שֶׁלֹּא יֹאמַר אָדָם לַחֲבֵרוֹ אַבָּא גָדוֹל מֵאָבִיךָ. וְשֶׁלֹּא יְהוּ מִינִין אוֹמְרִים, הַרְבֵּה רָשֻׁיּוֹת בַּשָּׁמָיִם. וּלְהַגִּיד גְּדֻלָּתוֹ שֶׁל הַקָּדוֹשׁ בָּרוּךְ הוּא, שֶׁאָדָם טוֹבֵעַ כַּמָּה מַטְבְּעוֹת בְּחוֹתָם אֶחָד וְכֻלָּן דּוֹמִין זֶה לָזֶה, וּמֶלֶךְ מַלְכֵי הַמְּלָכִים הַקָּדוֹשׁ בָּרוּךְ הוּא טָבַע כָּל אָדָם בְּחוֹתָמוֹ שֶׁל אָדָם הָרִאשׁוֹן וְאֵין אֶחָד מֵהֶן דּוֹמֶה לַחֲבֵרוֹ. לְפִיכָךְ כָּל אֶחָד וְאֶחָד חַיָּב לוֹמַר, בִּשְׁבִילִי נִבְרָא הָעוֹלָם. וְשֶׁמָּא תֹאמְרוּ מַה לָּנוּ וְלַצָּרָה הַזֹּאת, וַהֲלֹא כְבָר נֶאֱמַר (ויקרא ה) וְהוּא עֵד אוֹ רָאָה אוֹ יָדָע אִם לוֹא יַגִּיד וְגוֹ'. וְשֶׁמָּא תֹאמְרוּ מַה לָּנוּ לָחוּב בְּדָמוֹ שֶׁל זֶה, וַהֲלֹא כְבָר נֶאֱמַר (משלי יא) וּבַאֲבֹד רְשָׁעִים רִנָּה:
How does the court intimidate the witnesses in giving testimony for cases of capital law? They would bring the witnesses in and intimidate them by saying to them: Perhaps what you say in your testimony is based on conjecture, or perhaps it is based on a rumor, perhaps it is testimony based on hearsay, e.g., you heard a witness testify to this in a different court, or perhaps it is based on the statement of a trusted person. Perhaps you do not know that ultimately we examine you with inquiry and interrogation, and if you are lying, your lie will be discovered. The court tells them: You should know that cases of capital law are not like cases of monetary law. In cases of monetary law, a person who testifies falsely, causing money to be given to the wrong party, can give the money to the proper owner and his sin is atoned for. In cases of capital law, if one testifies falsely, the blood of the accused and the blood of his offspring that he did not merit to produce are ascribed to the witness’s testimony until eternity. The proof for this is as we found with Cain, who killed his brother, as it is stated concerning him: “The voice of your brother’s blood [demei] cries out to Me from the ground” (Genesis 4:10). The verse does not state: Your brother’s blood [dam], in the singular, but rather: “Your brother’s blood [demei],” in the plural. This serves to teach that the loss of both his brother’s blood and the blood of his brother’s offspring are ascribed to Cain. The mishna notes: Alternatively, the phrase “your brother’s blood [demei],” written in the plural, teaches that that his blood was not gathered in one place but was splattered on the trees and on the stones. The court tells the witnesses: Therefore, Adam the first man was created alone, to teach you that with regard to anyone who destroys one soul from the Jewish people, i.e., kills one Jew, the verse ascribes him blame as if he destroyed an entire world, as Adam was one person, from whom the population of an entire world came forth. And conversely, anyone who sustains one soul from the Jewish people, the verse ascribes him credit as if he sustained an entire world. The mishna cites another reason Adam the first man was created alone: And this was done due to the importance of maintaining peace among people, so that one person will not say to another: My father, i.e., progenitor, is greater than your father. And it was also so that the heretics who believe in multiple gods will not say: There are many authorities in Heaven, and each created a different person. And this serves to tell of the greatness of the Holy One, Blessed be He, as when a person stamps several coins with one seal, they are all similar to each other. But the supreme King of kings, the Holy One, Blessed be He, stamped all people with the seal of Adam the first man, as all of them are his offspring, and not one of them is similar to another. Therefore, since all humanity descends from one person, each and every person is obligated to say: The world was created for me, as one person can be the source of all humanity, and recognize the significance of his actions. The court says to the witnesses: And perhaps you will say: Why would we want this trouble? Perhaps it would be better not to testify at all. But be aware, as is it not already stated: “And he being a witness, whether he has seen or known, if he does not utter it, then he shall bear his iniquity” (Leviticus 5:1)? It is a transgression not to testify when one can do so. And perhaps you will say: Why would we want to be responsible for the blood of this person? But be aware, as is it not already stated: “When the wicked perish, there is song” (Proverbs 11:10)?