To determine for which equations [tex]\( x = 9 \)[/tex] is a possible solution, let's examine each equation step-by-step:
1. [tex]\( |x| = 9 \)[/tex]:
- The absolute value of [tex]\( x \)[/tex] is [tex]\( |9| = 9 \)[/tex].
- This holds true, so [tex]\( x = 9 \)[/tex] is a solution for this equation.
2. [tex]\( -|x| = 9 \)[/tex]:
- The absolute value of [tex]\( x \)[/tex] is [tex]\( |9| = 9 \)[/tex].
- Hence, [tex]\( -|9| = -9 \)[/tex], which does not equal 9.
- Therefore, [tex]\( x = 9 \)[/tex] is not a solution for this equation.
3. [tex]\( -|-x| = 9 \)[/tex]:
- The absolute value of [tex]\( -x \)[/tex] is [tex]\( |-9| = 9 \)[/tex].
- Thus, [tex]\( -|-9| = -9 \)[/tex], which does not equal 9.
- Hence, [tex]\( x = 9 \)[/tex] is not a solution for this equation.
4. [tex]\( -|-x| = -9 \)[/tex]:
- The absolute value of [tex]\( -x \)[/tex] is [tex]\( |-9| = 9 \)[/tex].
- Thus, [tex]\( -|-9| = -9 \)[/tex], which holds true.
- Therefore, [tex]\( x = 9 \)[/tex] is a solution for this equation.
5. [tex]\( |x| = -9 \)[/tex]:
- The absolute value [tex]\( |x| \)[/tex] is always non-negative.
- Therefore, [tex]\( |9| = 9 \)[/tex], and it can never be equal to -9.
- Thus, [tex]\( x = 9 \)[/tex] is not a solution for this equation.
6. [tex]\( |-x| = 9 \)[/tex]:
- The absolute value of [tex]\( -x \)[/tex] is [tex]\( |-9| = 9 \)[/tex].
- This holds true, so [tex]\( x = 9 \)[/tex] is a solution for this equation.
7. [tex]\( |-x| = -9 \)[/tex]:
- The absolute value [tex]\( |-x| \)[/tex] is always non-negative.
- Therefore, [tex]\( |-9| = 9 \)[/tex], and it can never be equal to -9.
- Hence, [tex]\( x = 9 \)[/tex] is not a solution for this equation.
Based on the step-by-step analysis, [tex]\( x = 9 \)[/tex] is a possible solution for the following equations:
- [tex]\( |x| = 9 \)[/tex]
- [tex]\( -|-x| = -9 \)[/tex]
- [tex]\( |-x| = 9 \)[/tex]
So, the equations for which [tex]\( x = 9 \)[/tex] is a possible solution are:
[tex]\[ |x|=9 \][/tex]
[tex]\[ -|-x|=-9 \][/tex]
[tex]\[ |-x|=9 \][/tex]
