Answer: Choice C (empty set)
The first thing to do is subtract 7 from both sides
|x| + 7 < 4
|x| + 7 - 7 < 4 - 7
|x| < -3
Recall that absolute value represents distance on a number line. Saying |x| is less than -3 means the distance is smaller than -3, which is impossible. Anything smaller than -3 is also negative. Distance can never be negative.
That leads us to the empty set as the solution. There are no values we can plug into any of the inequalities above, to have them lead to true statements.
Writing the special symbol
is one way to represent the empty set. Another way is to write curly braces only with nothing inside them like so: { }, so its a set with nothing inside it more or less.