The Modulus Function

The modulus of a number is its distance from zero, so $|-12| = 12$ and $|5| = 5$.

Both 9 and $-9$ are 9 units from zero. A modulus equation with a positive right side always has two solutions.

Either $x - 4 = 6$, giving $x = 10$, or $x - 4 = -6$, giving $x = -2$.

A 'less than' modulus gives one interval: $-8 \le x + 3 \le 8$, then subtract 3 throughout.

Either $3x - 2 > 7$, giving $x > 3$, or $3x - 2 < -7$, giving $3x < -5$ and $x < -\tfrac53$. Two separate intervals.

The parabola dips below the axis between $x = -2$ and $x = 2$; that section folds upwards, producing sharp corners at $(-2,0)$ and $(2,0)$. Everything above the axis is unchanged.