Smartest me plz how to solve this problem

(-∞, 4] U [-2, 1) U (3, ∞)

Step-by-step explanation:

I'm sure you look at the number line and in general know what's going on, but first, a few points:

a closed circle means the point is included

an open circle means all values around that point are included, but not that point exactly

an arrow means you assume the line goes on forever

So, with all that in mind, we can translate this number line into mathematical symbols.

It starts with an arrow pointing towards the negative values. Like we said, this means that the line goes on forever in that direction. In math symbols, this means that the arrow goes to -∞, and we can see a closed circle right over -4, which means that's where that interval stops. But do we use brackets [] or parentheses ()? The answer is related to the open and closed circles. For closed circles, they become closed intervals, which means they have to have brackets [ ]. Open circles represent open intervals, which use parentheses ( ). And finally, the arrows that go to positive or negative infinity are always contained using parentheses. This is because infinity is not a "touchable" value; we can't close an interval around infinity because infinity does not stop or end. It can't be contained in anything besides an open interval. So if we have something from -∞ to 4 with a closed circle, we write it in interval notation like (-∞, 4].

The rest of the intervals follow similarly. -2 has a closed circle, so it will need a bracket. -1 has an open circle, so it will need a parentheses, and we get [-2, 1). Finally, we have 3 with an open circle and then an arrow pointing to positive infinity. We write that like (3, ∞).

Now, to put those all together and use U for union, we just write

(-∞, 4] U [-2, 1) U (3, ∞)