Absolute value question false solution

Question

|x| = 3x – 2

Why does this statement eventually give you a solution that isn't valid.

So this equation comes out:

x = 3x - 2

2 = 2x

x = 1

OR

x = -3x + 2

4x = 2

x = 1/2

However 1/2 doesn't work. What's the rule here? How can you tell which absolute value questions have false solutions? What kinds of absolute value questions do not produce false solutions?

Answer 1

The equation $|x| = 3x - 2$ is equivalent to the two equations $x = 3x - 2$ when $x \geq 0$ and $-x = 3x - 2$ when $x < 0$. The second solution $x = 1/2$ is obtained using the assumption that $x<0$, and that is why it is not a solution, as obviously $1/2 > 0$ and is not considered for the equation.

Answer 2

You are using the following without explicitly stating it: "If $x<0$, then $|x|=-x$; if $x\geq 0$, then $|x|=x$." Combine the conditions with the solutions, and you will be able to eliminate the false solution because it does not satisfy the condition.