The number of distinct solutions of the equation |x-|2x+1||=3 is

1984 AMC 12 Problem 20

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20.

The number of distinct solutions of the equation

\left|x-|2x+1|\right|=3

0

1

2

3

4

Split the outer absolute value into two equations

For each equation, isolate

|2x+1|

before checking its two cases

x-|2x+1|=3,

then

|2x+1|=x-3,

which requires

x\ge3

and yields no solution. If

x-|2x+1|=-3,

then

|2x+1|=x+3.

Its two linear cases give

x=2

and

x=-\frac{4}{3},

both valid. Thus there are

2

distinct solutions.

Therefore, the correct answer is C.