|x| > 4

Square

⇒ x² > 16

⇒ x² – 16 > 0

⇒ x² – 4² > 0

⇒ (x + 4)(x – 4) > 0

Observe that when x is greater than 4, (x + 4)(x – 4) is positive

And for each root the sign changes hence

We want greater than 0 that is positive part

Hence x should be less than -4 and greater than 4 for (x + 4)(x – 4) to be positive

x less than -4 means x is from negative infinity to -4 and x greater than 4 means x is from 4 to infinity

Hence x ∈ (-∞, -4) and x ∈ (4, ∞)

Hence the solution set of |x| > 4 is (-∞, -4) U (4, ∞)