Find all numbers satisfying the given description. The

3/5 and -3

Pembahasan

Let the number be x.
The problem states that the absolute value of the number is the same as the absolute value of the difference of 1 and two-thirds of the number. We can write this as an equation:
|x| = |1 - (2/3)x|
This equation gives us two possibilities:
Case 1: x = 1 - (2/3)x
Add (2/3)x to both sides:
x + (2/3)x = 1
(3/3)x + (2/3)x = 1
(5/3)x = 1
Multiply both sides by 3/5:
x = 3/5
Let's check this solution: |3/5| = 3/5. |1 - (2/3)*(3/5)| = |1 - 6/15| = |1 - 2/5| = |3/5| = 3/5. So, x = 3/5 is a valid solution.
Case 2: x = -(1 - (2/3)x)
x = -1 + (2/3)x
Subtract (2/3)x from both sides:
x - (2/3)x = -1
(3/3)x - (2/3)x = -1
(1/3)x = -1
Multiply both sides by 3:
x = -3
Let's check this solution: |-3| = 3. |1 - (2/3)*(-3)| = |1 - (-2)| = |1 + 2| = |3| = 3. So, x = -3 is a valid solution.
Therefore, the numbers satisfying the description are 3/5 and -3.