Intervals
What are intervals good for in real life?
Although you don't realise it, you use intervals every day. Even your daily routine is one interval. You wake up at six in the morning and you go to bed at 22:00. This duration could be noted as the interval (6,22), while round brackets can be square because you don't know if you will get up exactly at six, or fall asleep exactly at twenty two. This logic leads us to a problem...
It doesn't matter in real life, if you get up exactly at six (if you get up at 6:01, nothing would happen, except you could miss the bus), but in mathematics, one must determin exactly when the person gets up, meaning that round brackets are very important. Look at the next examples encountered by almost every living person.
“Opening hours are from 9:00 to 18:00."
It means that you can come in during this time interval. Mathematically noted: [9,18],(9,18),[9,18) or (9,18], depending, if they close and open the shop at the exact times.
"Admission is free for children up to the age of 15."
Again, it depends if guests think that even children, who are exactly 15 years old can enter at no cost; thus, mathematically noted: (0,15] or (0,15). Another possibility is that if someone is 15 years and 5 months old, he/she is legally still 15; therefore, the discount should apply as well. This situation would be mathematically noted like this: (0,16).
"The weight limit of the platform is 1000\:\mathrm{kg.}"
Can the platform carry 1000 \mathrm{~kg} or a maximum of 999.99...kg? Mathematically, it has to be determined exactly, so either [0,1000] or [0, 1000). In real life, for example, the weight limit of a lift is much higher than stated because various physical must be considered. Thus, the real-life variant is [0, 1000], so don't worry that the lift wouldn't be able to carry you.
"No thoroughfare for vehicles with height over 3.5\mathrm{~m}.”
In this exercise it doesn't matter on the agreement of people, here it isn't reckoned with large excess (the sign says 3.5 \mathrm{~m}, so the actual height of the bridge is just a little bit higher, a few centimetres at the most); therefore, the interval here should use square brackets [0,3.5].
It always matters how people come to make an agreement, but in mathematics, it must always be clearly stated, and that's why brackets in intervals are so important: please, don't disregard it.