Absolute Value

y=\abs(x)

• properties:

  • $|-a|=|a|$

  • $|ab|=|a||b|$ (you can use this to find 1), $|\frac{a}{b}|=\frac{|a|}{|b|},b\ne 0$

  • $|a\pm b|\le |a|+|b|$ (triangle inequality)

  • $|a|<b⟺-b<a<b,b>0$

  • $|a|>b⟺a>b\text{ or }a<-b$

  • $\sqrt{a^2}=|a|$

• To solve, you must clearly define the cases (eg. $x\ge \frac{1}{2},x<\frac{1}{2}$)
  • find the zeros of all absolute parts
  • solve between the cases (make sure to include the edges somewhere)
  • Make sure the x solutions satisfy the restrictions!!