Does Ecclesiastes give an example of a time to tear and a time to mend? Ecclesiastes chapter 3 lists fourteen pairs of opposites, inluding "a time to tear and a time to mend". I have found one or more examples of most of the pairs elsewhere in the book, but can't find any for tearing or mending. For example, there are references to laughter, to mourning, to being born, to dying, to war and peace. I take chapter 3 as a sort of index into the rest of the book. For some of the words, I have to resort to emotional synonyms; there is no reference to dancing, but there are references to joy, and we dance when we are joyful. 

However, I can't find anything that I can match to clothing analogies
 like tearing and mending. Perhaps this is Solomon being silent about a
 touchy subject:


  10 Although he had forbidden Solomon to follow other gods, Solomon did
  not keep the Lord’s command. 11 So the Lord said to Solomon, “Since
  this is your attitude and you have not kept my covenant and my
  decrees, which I commanded you, I will most certainly tear the kingdom
  away from you and give it to one of your subordinates. 12
  Nevertheless, for the sake of David your father, I will not do it
  during your lifetime. I will tear it out of the hand of your son. 13
  Yet I will not tear the whole kingdom from him, but will give him one
  tribe for the sake of David my servant and for the sake of Jerusalem,
  which I have chosen.” (1 Kings 11:10-13)


Solomon makes a point of speaking about how unfair it is to leave your estate to someone who did not build it, and that person might not deserve it:


  18 I hated all the things I had toiled for under the sun, because I
  must leave them to the one who comes after me. 19 And who knows
  whether that person will be wise or foolish? Yet they will have
  control over all the fruit of my toil into which I have poured my
  effort and skill under the sun. This too is meaningless. (Ecclesiastes
  2:18-19)


Given all that, I would expect some additional reference to tearing in Ecclesiastes, but can't find it. Maybe the Hebrew admits some clothing references that the English does not show? Or maybe Solomon is "on the nose" for once and eschews a parable?
There are various ways of tackling the book of Ecclesiastes, with the example you give being a method I had never considered before; taking “chapter 3 as a sort of index into the rest of the book.” I know and appreciate the worth of taking an entire book of the Bible as needing to be read within its own context, detecting the writer’s style and paying attention to his repetitions, but that the rest of the Bible may also speak to the matter, giving further insights (either before that particular book was written, or after). I had never thought to examine chapter three’s 14 pairs of opposites, expecting to find additional reference to tearing (vs. 7) within Ecclesiastes.
Upon examination, the 14 couplets cover every range of human activity, beginning with life and death (vs. 2). Then come three creative and destructive activities (vss. 2-3) followed by human emotions both public and private (vs. 4).  A move is made to friendship and enmity. Apparently, in that era, stones refer to filling up your neighbour’s field with stones or gathering the stones they throw and using them to build something), and a time to say hello and a time to say goodbye (vs. 5). The next two have to do with possessions and our resolutions concerning them (vs. 6). Finally, a return is made to the various creative and destructive activities of man (vss. 7-8). The key point in all such affairs is the element of time – seasons – life’s rhythms. Ignore that and you suffer burn-out. But if we place our times in God’s hands, we will find balance and, at the end, God’s purpose which is detailed in the last chapter.
You referred to 1 Kings 11:10-13 as a possible connection. However, I do not know if God warned Solomon of tearing the kingdom away once it had been given to his son prior to Solomon collating all those wise sayings.
It seems likely that his collection was a life-time’s work with formation into a structured book happening after much consideration. Given how Solomon states the conclusion of the whole matter of the vanity of life in its last chapter 12, showing that it has purpose in obediently fearing God, who alone can break through the brassy heavens above us, I’ve viewed his last chapter as key to unlocking all that went before. And it speaks of the ultimate physical rending: the breakdown of the ageing body as death gets a grip as in 12:6-7:

“Or ever the silver cord be loosed, or the golden bowl be broken, or
the pitcher be broken at the fountain, or the wheel broken at the
cistern. Then shall the dust return to the earth as it was, and the
spirit shall return unto God who gave it.”

That is why 12:5 speaks of the mourners going about the streets. Old age and impending death have a way of focusing our minds, to expose vain thoughts and to seek God’s mind on matters to heal all the damage of a merely physical life.
Back to 3:7. One translation says, &quot;A time to rend, and a time to sew&quot;. [Y.L.T.] Rending was a term used in the Bible for showing great grief, when garments would be torn and sack-cloth worn instead, and ashes placed on the head. But the writer of Ecclesiastes, a rich powerful ruler, would never be engaged in sewing (repairing) such rent garments! However, he would be very involved in the showing of grief at times of mourning. Indeed, there would be times of national crisis when he would be expected to lead the nation in public demonstration of rending his fine clothes, to then wear sack-cloth and ashes.
That is why I think chapter 12 is a fitting link to 3:7. The whole chapter points to impending death, and when that time inevitably comes, it would be better for young people reading his wise sayings to have taken to heart its counsel, and to have sought God early. Only those who do that will no longer find the heavens as brass, “under the sun”, but know the satisfaction of living to honour God, before it comes time to die.