Solve.

|3w-1| = |3w|
or
abs(3w-1) - abs(3w)

I know you can use a calculator and find the answer easily, but we're supposed to know how to do this without a calculator.

four possibilites

A. (3w-1) = (3w)
B (3w-1) = -(3w)
C. -(3w-1) = (3w)
D. -3(w-1) = -(3w)
A and D are the same, use A
B and C are the same, use B
so
A:
3w-1 = 3w
0 * w = 1 no solution possible
B:
3 w - 1 = - 3 w
6 w = 1
w = 1/6

To solve the equation |3w - 1| = |3w|, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative.

Case 1: 3w - 1 ≥ 0 and 3w ≥ 0
In this case, we don't need to take the absolute value of the expressions. So the equation becomes:
3w - 1 = 3w

Simplifying the equation:
3w - 3w = 1
0 = 1

Since 0 ≠ 1, there are no solutions for this case.

Case 2: 3w - 1 < 0 and 3w < 0
In this case, we need to take the negative value inside the absolute value. So the equation becomes:
-(3w - 1) = - (3w)

Simplifying the equation by distributing the negative sign:
-3w + 1 = -3w

Now, we have the variable 'w' on both sides of the equation. By subtracting -3w from both sides, we get:
1 = 0

Again, 1 ≠ 0, so there are no solutions for this case either.

In summary, there are no values of 'w' that satisfy the equation |3w - 1| = |3w|. Therefore, the solution to the equation is an empty set, often represented as ∅ or {} in mathematics.