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?

the second is a solution only if $x\le 0$, there is no "false solution" — user127.0.0.1, Feb 08 '14 at 07:57
We (1) looked for solutions $\ge 0$ and (2) looked for solutions with $x\lt 0$. Found there are none of type (2). Nothing wrong with that. — André Nicolas, Feb 08 '14 at 08:03

score 2 · Accepted Answer · answered Feb 08 '14 at 08:02

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.

score 0 · Answer 2 · answered Feb 08 '14 at 08:03

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.

Absolute value question false solution

2 Answers2