Question

Solve for x if |x+2|=|2x+1| ?

Answer 1

`x = 1`

Explanation:

`x + 2 = 2x + 1`

Bring like terms together.

Subtract x from both sides,

`x + 2 -cancel x = cancel(2x)^color(red)x + 1 - cancelx`

`2 = x + 1`

Subtract 1 from both sides,

`cancel2^color(red)1 - cancel1 = x + cancel1 - cancel1`

`x = 1`

Answer 2

`x = pm 1`

Explanation:

`"We could square both sides : "`
`(x+2)^2 = (2x+1)^2`
`=> x^2 + cancel(4x) + 4 = 4x^2 + cancel(4x) + 1`
`=> 3x^2 - 3 = 0`
`=> x^2 = 1`
`=> x = pm 1`

`"The absolute value is > 0 and squaring yields also values > 0."`
`"So we will have the same solutions."`

`"We could also work with the definition of |x| : "`
`={(x " , "x>=0),(-x " , "x<=0):}`
`"But here we have 4 cases, 2 for the LHS (left hand side of"`
`"the equation) and 2 for the RHS, so it is a lot of work to deal"`
`"with 4 cases, squaring is easier."`