This is on my final exam review. I have the solution, but I do not understand it. When looking at $|x+1| - |3x - 1|$, I see four cases:
a) $x + 1 > 0$ and $3x - 1 > 0$
$x > -1$ and $x > \frac{1}{3}$
$x > -1$
b) $x + 1 < 0$ and $3x - 1 > 0$
$x < -1$ and $x > \frac{1}{3}$ (cannot happen)
c) $x + 1 > 0$ and $3x - 1 < 0$
$x > -1$ and $x < \frac{1}{3}$
$-1 < x < \frac{1}{3}$
d) $x + 1 < 0$ and $3x - 1 < 0$
$x < -1$ and $x < \frac{1}{3}$
$x < \frac{1}{3}$
So when solving the equation, I would look at cases a, c, and d. However, the solution says to use $x < -1$, $-1 \leq x < \frac{1}{3}$, and $\frac{1}{3} \leq x$
What am I missing here?