Question

Which number line represtents the solution to |x+4|=2?​

Answer 1

Points -2 and -6 on the number line are the two solutions.

Explanation:

Use the definition of absolute value as a starting point

`|x|=x\,\,\mbox{for}\,\,x\geq 0\\|x|=-x\,\,\mbox{for}\,\,x<0`

To solve the equation, you need to treat the two cases as above:

`|x+4|=x+4=2\,\,\,\mbox{for}\,\,x+4\geq 0\implies x\geq -4\\x+4=2\implies x=-2`

The solution x=-2 is consistent with the condition x>=-4, so it is the first and valid solution. Now the second case of the absolute value:

`|x+4|=-(x+4)=2\,\,\,\mbox{for}\,\,x+4< 0\implies x<-4\\-(x+4)=-x-4=2\implies x = -6`

Again, the second solution -6 complies with the requirement that x<-4, so it is valid.