difference between check and check out

check - (шалгах, шалгаж үзэх) we would not say "check out my bag". but we say "check my bag"  for example - "I will check that website" means "it probably means I am going back to the same site I have used before to see if there is an update, etc."
1 [transitive] to examine something to see if it is correct, safe, or acceptable
      ---- Check the container for cracks or leaks.
2 [intransitive, transitive] to find out if something is correct or true, or if something is how you think it is
       ---- “Is Mary in the office?” “Just a moment. I'll go and check.”
check out - (хянах, хянаж үзэх)  to be found to be true or acceptable after being examined - энэ өгүүлбэр нь бид ямар нэг шалган шинжилсэний дараа "хянаж and  хянаж үзэх" хийдэг. мөн энэ нь үүнтэй таарч байна - "Generally, check out" means to check/see/investigate for the first time."
1 to be found to be true or acceptable after being examined
      ---- The local police found her story didn't check out.
2 to take the things you want to buy at a supermarket to a counter and pay for them
      ---- As soon as I get a couple more items, I'll be ready to check out.

difference between check and check out

• Generally, "check out" means to check/see/investigate for the first time. 
  
  You would not "check out" your bag -- you have seen it many times, you just want "check" to be sure that you have everything.
  
  If I say I will "check out" a website, it means I will go see it for the first time. If I "check" it, it probably means I am going back to the same site I have used before to see if there is an update, etc.

• I will check/check out that website: to check a website is to look for something specific; to check it out is to take a look around in a general way. In practice there is a lot of overlap between the two.
  
  I have to check the soup and see if it's boiling.
  I want to check out that new store downtown and see what they sell.
  
  I am going to check/check out my bag to see if I have everything in it.
  
  We'd probably say "check my bag," although you have to be careful because "to check a bag" also means, for example, to leave it at a hotel desk or airport baggage area. Again:
  
  I'm going to check my bag and see if my sunglasses are in there.
  I'm going to check out the new bag they're selling in the store downtown.
  
  Hope that helps.