Why do some absolute value equations have solutions that aren’t correct?

Answer 1

Suppose the equation is this.....

|x| = f(x)

where x is some polynomial function, or even just a LINEAR function...

there are intervals where the function is NEGATIVE (below the x-axis).

However, the ABSOLUTE Value is NEVER NEVER negative...

Mechanically, you were taught to solve them, as per the following example..

Ex.

|x| = 2x+1

then x = 2x + 1 or x= -(2x+1)

0 = x+1 x = -2x-1

x = -1 3x = -1

x = -1/3

But when x = -1, the left side is 1 and the right side is -1, so it DOES NOT WORK!!

when x = -1/3, the left is 1/3 and the right is is 1/3, so it checks out...

therefore the only solution is x=-1/3

Now, considering the concept stated above |x| = f(x) = 2x+1

2x+1<0 when 2x < - 1 , which is when x < -1/2

So the equation has NO solutions when x<-1/2...

note that x= -1 < -1/2

SO whenever the equation has |x| = f(x) for ANY type of function, this Phenomena SHALL occur

unless the function is ALWAYS positive, which YOU have the burden to prove