$\text{a)}\left|x-3\right|<4$  

$x-3<4\text{i}x-3>-4$  

$x<7\text{i}x>-1$  

$x\in \left(-1,7\right)$  

---

$\text{b)}\left|x+1\right|<3$  

$x+1<3\text{i}x+1>-3$  

$x<2\text{i}x>-4$  

$x\in \left(-4,2\right)$  

---

$\text{c)}\left|x+2\right|>2$  

$x+2>2\text{lub}x+2<-2$  

$x>0\text{lub}x<-4$  

$x\in \left(-\infty ,-4\right)\cup \left(0,+\infty \right)$  

---

$\text{d)}\left|x-4\right|\ge 3$  

$x-4\ge 3\text{lub}x-4\le -3$  

$x\ge 7\text{lub}x\le 1$  

$x\in (-\infty ,1⟩\cup ⟨7,+\infty )$  

---

$\text{e)}\left|x+1\right|>1$  

$x+1>1\text{lub}x+1<-1$  

$x>0\text{lub}x<-2$  

$x\in \left(-\infty ,-2\right)\cup \left(0,+\infty \right)$  

---

$\text{f)}\left|x+1\right|\le \sqrt{2}$  

$x+1\le \sqrt{2}\text{i}x+1\ge -\sqrt{2}$  

$x\le \sqrt{2}-1\text{i}x\ge -\sqrt{2}-1$  

$x\in ⟨-\sqrt{2}-1,\sqrt{2}-1⟩$