Enter (without the quotation marks) "$==>$" if the left equation implies the right, "$<==$" if the right equation implies the left, "$<==>$" if either equation implies the other, and "$><$" if neither equation implies the other.

$|x+2|=|1-2x|$ $x+2 = 1-2x$.

$x+2 = 1-2x$ $x=-\frac{1}{3}$.

$|x+2|=|1-2x|$ $-(x+2) = 1-2x$.

$-(x+2) = 1-2x$ $x=3$.
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The above considerations suggest that the original equation $$|x+2| = |1-2x|$$ has the two solutions $x=3$ and $x=-\frac{1}{3}$. Substituting in the original equation shows that this is indeed the case.