What does this notation imply: min {x, 5-x} ?

Answer 1

min{a,b} equals the minimum value chosen between a and b. This generalizes to min{a1,a2,...,an}, which is the minimum of all the values. For example:

min{3,5} = 3
min{5,3} = 3
min{1,5,8,3,55,7874,12} = 1

So, min{x,5-x} is the least value. When is x the lower of the two?

x < 5-x
if and only if
2x < 5
if and only if
x < 5/2

When is 5-x the lower of the two?

5-x < x
if and only if
5 < 2x
if and only if
5/2 < x

When x = 5/2, then 5-x = 5-(5/2) = 10/2 - 5/2 = 5/2, and so they are the same.

Therefore:
min{x,5-x} = x if x < 5/2 or 5-x if x >= 5/2.

Hope this helps!

Answer 2

min is short for minimum, and means the lowest element of the set. So min {x, 5-x} will be x when x ≤ 5-x, i.e. x ≤ 5/2, and it will be 5-x when x ≥ 5-x, i.e. x ≥ 5/2.

If you graph x and 5-x you will see that they are straight (and perpendicular) lines crossing at x = 5/2. On the left of 5/2 x is smaller, and on the right of 5/2 5-x is smaller. So we pick the lower line in each half of the graph.

Answer 3

The lesser (minimum) of the two element set comprised of the value "x", and the value "5-x".

For integer values:
Where x < 3, the first element ("x") would be the lesser number.
Where x >=3, the second element ("5-x") would be the lesser number.