Let's analyze each of the given equations to determine if [tex]\( x = -3 \)[/tex] is a solution.
1. Equation: [tex]\(|x| = 3\)[/tex]
To solve [tex]\(|x| = 3\)[/tex], compute the absolute value of [tex]\( x \)[/tex]:
[tex]\[
|x| = |-3| = 3
\][/tex]
Since [tex]\( 3 = 3 \)[/tex], this equation is true.
2. Equation: [tex]\(|x| = -3\)[/tex]
The absolute value of any number is always non-negative (i.e., greater than or equal to zero). Therefore, [tex]\(|x| = -3\)[/tex] doesn't make sense because [tex]\(-3\)[/tex] is negative. This equation is false.
3. Equation: [tex]\(|-x| = 3\)[/tex]
Compute the absolute value of [tex]\(-x\)[/tex] where [tex]\( x = -3 \)[/tex]:
[tex]\[
|-x| = -(-3) = |-3| = 3
\][/tex]
Since [tex]\( 3 = 3 \)[/tex], this equation is true.
4. Equation: [tex]\(|-x| = -3\)[/tex]
As previously mentioned, the absolute value must always be non-negative. Therefore, [tex]\(|-x| = -3\)[/tex] does not make sense and is false.
5. Equation: [tex]\(-|x| = -3\)[/tex]
Compute the absolute value of [tex]\( x \)[/tex] and then take the negative of it:
[tex]\[
-|x| = -|-3| = -3
\][/tex]
Since [tex]\(-3 = -3\)[/tex], this equation is true.
Based on the analysis, the three equations for which [tex]\( x = -3 \)[/tex] is a solution are:
1. [tex]\(|x| = 3\)[/tex]
2. [tex]\(|-x| = 3\)[/tex]
3. [tex]\(-|x| = -3\)[/tex]
