Why do people suffer has been answered here.
I would summarise it by saying Muslims believe God tries and tests people in different ways, through both hardship and ease, as part of the "grand exam" of life.
We made a primordial agreement with God that we would submit ourselves fully to him and serve him with the aim of pleasing him with the consequence of eternal reward or punishment based on that - our life here seeks to test whether we are genuine in that or not.
While the cause of suffering is not always evident, the way that people respond to difficulty is their test.
Responding to hardship with patience and fortitude is a virtue for which Muslims believe a great reward is promised in this life and the afterlife.
Additionally, there is often "a silver lining" behind difficulties, whether we perceive it or not in the short term.
For instance, major disasters often bring out the best in people, inspiring them to perform remarkable acts as they respond to a personal or another's hardship with compassion and courage.
I would cite these points in addition to what has been written here:
1) We are his slaves and He is our lord who has total sovereignty over us.
"He is Allah , other than whom there is no deity, the Sovereign, the Pure, the Perfection, the Bestower of Faith, the Overseer, the Exalted in Might, the Compeller, the Superior. Exalted is Allah above whatever they associate with Him." [Surah al-Hashr, verse 23]
2) To believe in Him and worship Him via every aspect of our lives.
"Say. Surely my prayer and my sacrifice and my life and my death are (all) for Allah, the Lord of the worlds;" [Surah al-Anam, verse 162]
3) A means to distinguish us by testing our belief and efforts to please him.
"Do the people think they will be left to say "we believe" and that they won't be tested? But we have certainly tested those before them and we will certainly make evident those who are truthful and those who are liars." [Surah al-Ankabut, verses 2 and 3]
The point is we are here to worship our creator who has every right to put us through anything He pleases since He determines those rights in the first place, he determines what our souls can be put through within this life (which by the way is pretty insignificant when you think about it) and he determines what love means on the back of that relationship.
Aside from this when you believe in an afterlife and the concept of heaven and hell, the suffering is especially seen as a means to an end rather than the be all and end all.
There is another point for suffering that gets brought about via people as well as to why people have the ability to cause suffering.
"Verily Allah does not change the condition of a people until they change what is within themselves" [Surah ar-Rad, verse 11]
In above verse Allah (swt) is referring to a society that adopts an ideology and acts upon it, where Allah (swt) has written for the condition of that nation to vary according to the ideology that the society adopts. I.e. societies that adopt the Islamic ideology which is the truth will have less oppression etc... whereas societies that adopt the incorrect ideology which arose from limited minds of man will give rise to corruption and oppression due to the incapacity to produce a system that caters for our instincts.
So Allah (swt) has written for people to experience suffering in light of these two concepts. We're accountable for our approach to this suffering, how it was brought about and how we dealt with it.
On the back of that, to me, the fact that we suffer gets put into perspective and actually makes sense.
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