Question

Solve for `x` in `|2x-4| >= -5`? Why doesn't the usual method work in this case?

At first I was blindly solving and got `-0.5<=x<=4.5`. However, the answer is obviously all reals because the absolute value of anything will be positive; therefore, `abs(2x-4)` must be greater than `-5`.

Why doesn't the usual method work in this case?

Answer 1

`|2x-4| >= -5`

Since all modulus values are bigger or equal to `0`,
`|2x-4| >= 0`

Square both sides which gets rid of the modulus function,
`4x^2-16x+16>=0`
`(x-2)^2>=0`

`x>=2 or x<=2`

Hence, the solution is all real roots.

All absolute values must be equal or bigger to `0`, and hence, all values of `x` will work.

So, why doesn't the usual method work?
That is because we normally do this:

`|2x-4| >= -5`

Square both sides which gets rid of the modulus function,
`4x^2-16x+16>=25`
`4x^2-16x-9>=0`
`(2x-9)(2x+1)>=0`
`x<=-0.5` or `x>=4.5`

This is because we squared a negative number to make it positive, where in fact, is impossible as all absolute values are positive. Hence, the equation automatically implies that `25` is `5^2` instead of `(-5)^2`, resulting in the solution being (`x<=-0.5` or `x>=4.5`) instead of infinite number of solutions.