Sure, let's solve the inequality [tex]\( |x - 6| \geq 5 \)[/tex].
The absolute value inequality [tex]\( |x - 6| \geq 5 \)[/tex] can be broken into two separate inequalities:
1. [tex]\( x - 6 \geq 5 \)[/tex]
2. [tex]\( -(x - 6) \geq 5 \)[/tex] or equivalently [tex]\( x - 6 \leq -5 \)[/tex]
Now let's solve each inequality separately.
### Solving the first inequality:
[tex]\[ x - 6 \geq 5 \][/tex]
To isolate [tex]\( x \)[/tex], add 6 to both sides:
[tex]\[ x \geq 11 \][/tex]
### Solving the second inequality:
[tex]\[ x - 6 \leq -5 \][/tex]
To isolate [tex]\( x \)[/tex], add 6 to both sides:
[tex]\[ x \leq 1 \][/tex]
### Combining the solutions:
The two solutions to the inequality [tex]\( |x - 6| \geq 5 \)[/tex] are:
[tex]\[ x \geq 11 \][/tex]
or
[tex]\[ x \leq 1 \][/tex]
Thus, the complete solution combines these two cases. Therefore, the solution to the inequality [tex]\( |x - 6| \geq 5 \)[/tex] is:
[tex]\[ x \geq 11 \text{ or } x \leq 1 \][/tex]
The correct answer is:
C. [tex]\( x \geq 11 \text{ or } x \leq 1 \)[/tex]
