How do you solve $|x + 6| = 2x$ and find any extraneous solutions?

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Answer 1 · 2016-06-16T19:34:26

$x=-2$ if $x<-6$
$x=6$ if $x>=-6$

Knowing the property of absolute value that says:

if $|x|=a$ then
$x=-a , x<0$

$x=a,x>=0$

Applying the above property we have:

$x+6=-2x,x+6<0$ . Eq(1)

$x+6=2x,x+6>=0$ . Eq(2)

Solving the above equations we have:

Eq(1): if $x+6<0$ that is $x<-6$
$x+2x=-6$
$3x=-6$
$x=-6/3=-2$
So,
$x=-2,x<-6$

Eq(2): if $x+6>=0$ that is $x>=-6$

$x+6=2x$
$x-2x=-6$
$-x=-6$
$x=6$ whenever $x>=-6$

How do you solve |x + 6| = 2x|x + 6| = 2x and find any extraneous solutions?