To solve the compound absolute value inequality [tex]\( 3 \leq |x+2| \leq 6 \)[/tex], it is essential to consider the properties and implications of absolute value.
The absolute value inequality [tex]\( |x+2| \leq 6 \)[/tex] can be separated into:
1. [tex]\( x+2 \leq 6 \)[/tex]
2. [tex]\( x+2 \geq -6 \)[/tex]
The absolute value inequality [tex]\( |x+2| \geq 3 \)[/tex] can be separated into:
1. [tex]\( x+2 \leq -3 \)[/tex]
2. [tex]\( x+2 \geq 3 \)[/tex]
So, the inequalities that can be used to determine the solution to the given absolute value inequality are:
- [tex]\( x+2 \leq 6 \)[/tex]
- [tex]\( x+2 \geq -6 \)[/tex]
- [tex]\( x+2 \leq -3 \)[/tex]
- [tex]\( x+2 \geq 3 \)[/tex]
Hence, the correct answers are:
- [tex]\( x+2 \leq 6 \)[/tex]
- [tex]\( x+2 \geq -6 \)[/tex]
- [tex]\( x+2 \leq -3 \)[/tex]
- [tex]\( x+2 \geq 3 \)[/tex]
