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 values of x. - x = |x|
since |x| is always positive, you want x≤0
Well, isn't this equation a bit of a clown act? Let's break it down, shall we?
We have the equation -x = |x|.
Now, when we take the absolute value of a number, it throws away any negative signs and gives us the positive version (or zero if the number itself is zero). So let's consider two cases:
Case 1: x is positive or zero.
In this case, |x| is equal to x, since x itself is already positive or zero. So, we have -x = x. To solve this equation, we can add x to both sides, giving us 0 = 2x. Dividing both sides by 2, we find x = 0.
Case 2: x is negative.
In this case, |x| is equal to -x, as it gets rid of the negative sign. So, we have -x = -x. Well, isn't that a funny coincidence? This equation is true for all x that are negative.
So, putting it all together, we have that x = 0 or x is negative. On a number line, that would look like this:
<---o-----------------------------------------o--->
-∞ ... -3 -2 -1 0 1 2 3 ... ∞
The points to the left of zero (including zero itself) are the values of x that satisfy the equation -x = |x|.
To determine the values of x for which the algebraic statement -x = |x| is true, we can break it down into two cases and solve each separately.
Case 1: x is positive or zero
In this case, the absolute value of x is equal to x itself. Therefore, we can rewrite the equation as -x = x.
Adding x to both sides, we get: 0 = 2x
Dividing both sides by 2, we find: x = 0
Case 2: x is negative
In this case, the absolute value of x is equal to -x. Therefore, we can rewrite the equation as -x = -x.
This equation is true for all x values in the set of negative numbers.
Combining both cases, we have two solutions:
1. x = 0
2. x is any negative number (x < 0)
To represent these values on a number line:
```
--------------------|---|---|---|---|---|---|---|---|
... -4 -3 -2 -1 0 1 2 3 4 ...
```
On the number line, the set of points representing the values of x would be all the negative numbers and 0.
To solve the equation -x = |x|, we need to determine the values of x that satisfy the equation. Let's break it down step by step:
Step 1: Understanding the Absolute Value
The absolute value of a number, denoted by |x|, is the distance of that number from zero on the number line. It is always a non-negative value.
Step 2: Splitting into Two Cases
Since we have the absolute value of x on the right side of the equation, we need to consider two cases: when x is positive and when x is negative.
Case 1: x is positive
If x is positive, then |x| is equal to x. Therefore, we can rewrite the equation for this case as: -x = x.
Case 2: x is negative
If x is negative, then |x| is equal to -x (since for negative x, the absolute value of x is its negation). Therefore, we can rewrite the equation for this case as: -x = -x.
Step 3: Solving the Equations
Now let's solve each equation from the two cases separately:
Case 1: -x = x
To solve this equation, we want to isolate x on one side. Adding x to both sides of the equation gives us: -x + x = x + x
Simplifying, we get: 0 = 2x
Dividing both sides by 2, we obtain: 0/2 = x
Therefore, x = 0.
Case 2: -x = -x
In this case, the equation is already balanced and true for any value of x.
Step 4: Representing the Solutions on a Number Line
To show the set of points corresponding to the values of x, we can represent them on a number line:
--+-----+-----+-----+-----+-----
-2 -1 0 1 2
Since the equation -x = |x| is true for x = 0 in Case 1, we mark 0 on the number line.
The solution set, therefore, consists of only one value: x = 0.
On the number line, this is represented by a single point at 0.
I hope this clarifies how to determine the values of x for which the given algebraic statement is true and how to represent them on a number line. Let me know if you have any further questions!