Which equation does NOT have the set of all numbers as its solution?

1. |-x| = |x|

2. ||x|| = |x|

3. |-x| = -(-(-x))

4. -(-x) = x

I need help with this problem.

3.

Becouse :

| - x | = x

- ( - ( - x ) ) = - x

| - x | = - ( - ( - x ) ) mean:

x = - x

That is true only for x = 0

Actually, since -(-(-x)) = -x

we have
|-x| = -x

which is true for all x <= 0

Still the correct answer, though.

To determine which equation does NOT have the set of all numbers as its solution, we can systematically go through each equation and check if it satisfies the condition of having all numbers as solutions.

1. |-x| = |x|
To simplify this equation, we can consider the two possibilities for the absolute value of any number x:
- If x is positive or zero, the equation simplifies to x = x, which is always true.
- If x is negative, the equation simplifies to -x = x, which is not true for any negative number x.
Therefore, this equation does NOT have the set of all numbers as its solution.

2. ||x|| = |x|
In this equation, the double absolute value implies that the expression inside the inner absolute value must be non-negative. Since the absolute value of any number is always non-negative, this equation holds for all real numbers. So, this equation DOES have the set of all numbers as its solution.

3. |-x| = -(-(-x))
To simplify this equation, we can take x = 1 as an example. Plugging in 1, the equation becomes |-1| = -(-(-1)). Simplifying further, we get 1 = -(-1), which is not true. Hence, this equation does NOT have the set of all numbers as its solution.

4. -(-x) = x
This equation simplifies to x = x, which is true for all real numbers. Therefore, this equation DOES have the set of all numbers as its solution.

In conclusion, out of the given equations, the equation that does NOT have the set of all numbers as its solution is equation 1, |-x| = |x|.