What is the vertex of the quadratic function?
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When you represent intervals on the number line, you're including full dots, excluding empty dots, and you're considering numbers highlighted by the line.
In the first case, you've highlighted everything before -2 (full dot, thus included), and everything after 1 (empty dot, excluded). So, the set would be
or, in interval notation,
In the second case, you are looking for all numbers between -3 and 5. This interval is symmetric with respect to 1: you're considering all numbers that are at most 4 units away from 1, both to the left and to the right.
This means that the difference between your numbers at 1 must be at most 4, which is modelled by
where the absolute values guarantees that you'll pick numbers to the left and to the right of 1.