Consider the following algebraic statements and determine the values of x for which each statement is true. On a number line, show the set of all points corresponding to the value of x: x=|-x|

|x| is always ≥ 0

so, you need x≥0

Can u answer the same question but with [3x]-6

pretend the absolute value lines are the parentheses

Ah, the absolute value of x equals the absolute value of negative x! This is an interesting one. Let's solve it, but don't worry, I won't give you a negative answer!

Now, the absolute value of a number is always non-negative. So, we can split this equation into two cases:

Case 1: x is non-negative (x ≥ 0)
In this case, |-x| simplifies to just x itself. So the equation becomes x = |x|.

Case 2: x is negative (x < 0)
Now, |-x| simplifies to -x (because the absolute value of a negative number is the positive version of that number). So the equation becomes x = |-x| is equivalent to x = -x.

Now let's put these cases together on a number line:

____________________________________
| | | |
-x 0 x 2x

In Case 1 (x ≥ 0), the solution is all the non-negative values from 0 to infinity.
In Case 2 (x < 0), the solution is just x = 0, since that's the only value where x equals its negation.

So, putting it all together, the set of all points corresponding to the value of x is:

{x ∈ ℝ | x ≥ 0} ⋃ {0}

To determine the values of x for which the equation x = |-x| is true, we can break it down into two cases:

Case 1: x is positive or zero
In this case, |-x| is equal to x itself because the absolute value of a positive or zero number is the number itself. Therefore, the equation becomes x = x. This is true for all positive or zero values of x.

Case 2: x is negative
In this case, |-x| is equal to -x because the absolute value of a negative number is its positive counterpart. Therefore, the equation becomes x = -x. To solve for x, we can move all terms containing x to one side of the equation:

x + x = 0
2x = 0
x = 0

So, the value of x for which the equation x = |-x| is true is x = 0.

On a number line, we can represent the set of all points corresponding to the value of x={0} as a single point located at zero.

To determine the values of x for which the algebraic statement x = |-x| is true, we need to solve the equation.

First, let's simplify the absolute value expression |-x| by considering two cases: when x is positive and when x is negative.

When x is positive (x > 0), |-x| becomes -x since the absolute value of a positive number is the number itself with no change in sign.

So, the equation becomes x = -x.

Next, when x is negative (x < 0), |-x| becomes x since the absolute value of a negative number is the number itself with the sign flipped.

In this case, the equation becomes x = x.

Let's solve these two cases separately:

Case 1: x = -x

Adding x to both sides: x + x = 0
Combining like terms: 2x = 0
Dividing both sides by 2: x = 0

Case 2: x = x

This equation is true for all values of x since both sides are equal for any x.

Now, let's plot the values of x on a number line:

-∞ 0 ∞

The set of all points corresponding to the value of x is just the number zero (0) since x = |-x| is only true when x = 0.

So, on the number line, we would mark just the point zero (0).