Solve for x: |2x+|x|+1|=3
This appears to me this says: the absolute value of the sum of 2x, the absolute value of x, and 1 equals 3. Which might otherwise be written as: Abs(2x + Abs(x) + 1) = 3. If I'm correct, then to solve it do the following:
First, with any absolute value problem, we can break the (outer) absolute value up into a positive and negative version of the equation:
The equation |2x + |x| + 1| = 3 becomes:
2x + |x| + 1 = 3
AND
2x + |x| + 1 = -3
Solve 2x + |x| + 1 = 3 for |x|:
[1] |x| = -2x + 2
Solve 2x + |x| + 1 = -3 for |x|:
[2] |x| = -2x - 4
Now we need to break up equations [1] and [2] into their positive and negative versions:
Equation [1], |x| = -2x + 2, becomes:
[1a] x = -2x + 2
AND
[1b] x = -(-2x + 2) = 2x - 2
Equation [2], |x| = -2x - 4, becomes:
[2a] x = -2x - 4
AND
[2b] x = -(-2x - 4) = 2x + 4
From all of that, we now have 4 equations to solve:
Solve [1a], x = -2x + 2, for x:
3x = 2
x = 2/3
Solve [1b], x = 2x - 2, for x:
-x = -2
x = 2
Solve [2a], x = -2x - 4, for x:
3x = -4
x = -4/3
Solve [2b], x = 2x + 4, for x:
-x = 4
x = -4
Finally, check each answer in the original problem to see which solutions work. I'll leave that part for you to do. But in case you don't know how to, just plug in each x value: x = 2/3, 2, -4/3, and -4, into the original equation: |2x + |x| + 1| = 3