To determine for which equations [tex]$x = -3$[/tex] is a solution, let's analyze each equation step-by-step:
1. Equation: [tex]\( |x| = 3 \)[/tex]
- Substitute [tex]\( x = -3 \)[/tex] into the equation: [tex]\( |-3| \)[/tex]
- The absolute value of [tex]\(-3\)[/tex] is [tex]\( 3 \)[/tex]: [tex]\( |-3| = 3 \)[/tex]
- Hence, this equation holds true when [tex]\( x = -3 \)[/tex].
2. Equation: [tex]\( |x| = -3 \)[/tex]
- Substitute [tex]\( x = -3 \)[/tex] into the equation: [tex]\( |-3| \)[/tex]
- The absolute value of [tex]\(-3\)[/tex] is [tex]\( 3 \)[/tex]: [tex]\( |-3| = 3 \)[/tex]
- Since an absolute value cannot be negative, [tex]\( |x| \)[/tex] can never equal [tex]\(-3\)[/tex].
- Therefore, this equation cannot hold true for any [tex]\( x \)[/tex], including [tex]\( x = -3 \)[/tex].
3. Equation: [tex]\( |-x| = 3 \)[/tex]
- Substitute [tex]\( x = -3 \)[/tex] into the equation: [tex]\( |-(-3)| \)[/tex]
- Simplify the expression: [tex]\( |3| \)[/tex]
- The absolute value of [tex]\( 3 \)[/tex] is [tex]\( 3 \)[/tex]: [tex]\( |3| = 3 \)[/tex]
- Therefore, this equation holds true when [tex]\( x = -3 \)[/tex].
4. Equation: [tex]\( |-x| = -3 \)[/tex]
- Substitute [tex]\( x = -3 \)[/tex] into the equation: [tex]\( |-(-3)| \)[/tex]
- Simplify the expression: [tex]\( |3| \)[/tex]
- The absolute value of [tex]\( 3 \)[/tex] is [tex]\( 3 \)[/tex]: [tex]\( |3| = 3 \)[/tex]
- Since an absolute value cannot be negative, [tex]\( |-x| \)[/tex] can never equal [tex]\(-3\)[/tex].
- Hence, this equation cannot hold true for any [tex]\( x \)[/tex], including [tex]\( x = -3 \)[/tex].
5. Equation: [tex]\( -|x| = -3 \)[/tex]
- Substitute [tex]\( x = -3 \)[/tex] into the equation: [tex]\( -|-3| \)[/tex]
- Simplify the expression: [tex]\( -3 \)[/tex]
- Since [tex]\( -3 \)[/tex] equals [tex]\( -3 \)[/tex], the equation holds true when [tex]\( x = -3 \)[/tex].
Therefore, the three equations for which [tex]\( x = -3 \)[/tex] is a solution are:
1. [tex]\( |x| = 3 \)[/tex]
2. [tex]\( |-x| = 3 \)[/tex]
5. [tex]\( -|x| = -3 \)[/tex]
Thus, the correct choices are:
1. [tex]\( |x| = 3 \)[/tex]
3. [tex]\( |-x| = 3 \)[/tex]
5. [tex]\( -|x| = -3 \)[/tex]
