Question

Solve the inequality:|x-2| +x>=0

Answer 1

Following the definition of absolute value:

`|a|=\mleft\{\begin{aligned}\text{a if a}\ge0 \\ -a\text{ if a<0}\end{aligned}\mright.`

Applying it in this case we have:

`|x-2|=\mleft\{\begin{aligned}x-2\text{ if x-2}\ge0 \\ -(x-2)\text{ if x-2<0}\end{aligned}\mright.`

For the first case, we have the following:

`\begin{gathered} x-2\ge0 \\ \Rightarrow x-2+x\ge0 \\ \Rightarrow2x\ge2 \\ \Rightarrow x\ge(2)/(2)=1 \\ x\ge1 \end{gathered}`

For the second case, we have:

`\begin{gathered} -(x-2)+x\ge0 \\ \Rightarrow-x+2+x\ge0 \\ \Rightarrow2\ge0 \end{gathered}`

Which is true, therefore, the solution is x >= 1