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Unanswered Prayers
"I recognize that, on occasion, some of our most fervent prayers may seem to go unanswered. We wonder, 'Why?' I know that feeling! I know the fears and tears of such moments. But I also know that our prayers are never ignored. Our faith is never unappreciated. I know that an all-wise Heavenly Father's perspective is much broader than is ours. While we know of our mortal problems and pain, He knows of our immortal progress and potential. If we pray to know His will and submit ourselves to it with patience and courage, heavenly healing can take place in His own way and time."
Ref. Russell M. Nelson, "Jesus Christ--the Master Healer," Ensign, Nov. 2005, 86
I know that my prayers are always answered. They just aren't always answered in the manner I would like. Sometimes the answer is no or not now. That can seem like they just aren't being answered at all, but in reality they are.
The "catch" with prayers is that we should have the Spirit always, which will tell us what to pray for! In which case, presumably our prayers will always be acceptable and listened to and answered.
Haven't you ever had the experience of really wanting something, yet for some reason you're unable to ask the Lord to give it to you? I have. Maybe it's the Spirit telling you not to pray for it.
One time I prayed very hard that I would get a job I had applied for, but I didn't get the job. I was really, really crushed that the Lord hadn't helped me get that job, which had been my goal for years. It wasn't until years later that I realized I wasn't actually capable of doing that job well, and would probably have lost it, humiliatingly, if I had been hired. So it was in fact a blessing to me not to have gotten that job. I should have trusted the Lord's goodness and mercy at the time, rather than feeling picked on and whining, "Why, why, why?"
Now I'm more humble, and careful to say "Thy will be done" when I pray for things. What if I got what I prayed for, and it turned out to be horrible for me?