To determine for which equations [tex]\( x = -3 \)[/tex] is a possible solution, let's analyze each equation step-by-step.
1. Equation: [tex]\(|x| = 3\)[/tex]
The absolute value of [tex]\( x \)[/tex] is defined as:
[tex]\[
|x| = \begin{cases}
x, & \text{if } x \geq 0 \\
-x, & \text{if } x < 0
\end{cases}
\][/tex]
For [tex]\( x = -3 \)[/tex]:
[tex]\[
|x| = |-3| = 3
\][/tex]
Therefore, [tex]\( x = -3 \)[/tex] satisfies the equation [tex]\(|x| = 3\)[/tex].
2. Equation: [tex]\(|x| = -3\)[/tex]
The absolute value of any number is always non-negative, meaning [tex]\(|x| \geq 0\)[/tex]. Therefore, there is no value of [tex]\( x \)[/tex] that can make [tex]\(|x| = -3\)[/tex].
Hence, [tex]\( x = -3 \)[/tex] does not satisfy the equation [tex]\(|x| = -3\)[/tex].
3. Equation: [tex]\(|-x| = 3\)[/tex]
The expression [tex]\(-x\)[/tex] is the negation of [tex]\( x \)[/tex]. For [tex]\( x = -3 \)[/tex], we have [tex]\(-x = 3\)[/tex].
Now, find the absolute value:
[tex]\[
|-x| = |3| = 3
\][/tex]
Therefore, [tex]\( x = -3 \)[/tex] satisfies the equation [tex]\(|-x| = 3\)[/tex].
4. Equation: [tex]\(|-x| = -3\)[/tex]
Similar to equation 2, [tex]\(|-x|\)[/tex] represents the absolute value of [tex]\(-x\)[/tex], which must be non-negative. Thus, [tex]\(|-x| = 3\)[/tex] but cannot equal [tex]\(-3\)[/tex].
Hence, [tex]\( x = -3 \)[/tex] does not satisfy the equation [tex]\(|-x| = -3\)[/tex].
5. Equation: [tex]\(-|x| = -3\)[/tex]
First, determine [tex]\(|x|\)[/tex] for [tex]\( x = -3 \)[/tex]:
[tex]\[
|x| = |-3| = 3
\][/tex]
Then, negate this value:
[tex]\[
-|x| = -3
\][/tex]
Therefore, [tex]\( x = -3 \)[/tex] satisfies the equation [tex]\(-|x| = -3\)[/tex].
In summary, [tex]\( x = -3 \)[/tex] is a possible solution for the following three equations:
1. [tex]\(|x| = 3\)[/tex]
2. [tex]\(|-x| = 3\)[/tex]
3. [tex]\(-|x| = -3\)[/tex]
Thus, the correct options are:
- [tex]\(|x| = 3\)[/tex]
- [tex]\(|-x| = 3\)[/tex]
- [tex]\(-|x| = -3\)[/tex]
