Following the definition of absolute value:
[tex]|a|=\mleft\{\begin{aligned}\text{a if a}\ge0 \\ -a\text{ if a<0}\end{aligned}\mright.[/tex]Applying it in this case we have:
[tex]|x-2|=\mleft\{\begin{aligned}x-2\text{ if x-2}\ge0 \\ -(x-2)\text{ if x-2<0}\end{aligned}\mright.[/tex]For the first case, we have the following:
[tex]\begin{gathered} x-2\ge0 \\ \Rightarrow x-2+x\ge0 \\ \Rightarrow2x\ge2 \\ \Rightarrow x\ge\frac{2}{2}=1 \\ x\ge1 \end{gathered}[/tex]For the second case, we have:
[tex]\begin{gathered} -(x-2)+x\ge0 \\ \Rightarrow-x+2+x\ge0 \\ \Rightarrow2\ge0 \end{gathered}[/tex]Which is true, therefore, the solution is x >= 1
