|x| is always β₯ 0
so, you need xβ₯0
so, you need xβ₯0
pretend the absolute value lines are the parentheses
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 β₯ 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:
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-x 0 x 2x
In Case 1 (x β₯ 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 β β | x β₯ 0} β {0}
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.
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:
-β 0 β
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).