To find the set of [tex]\( x \)[/tex] values that satisfy the inequality [tex]\( |x| - 4 \geq 10 \)[/tex], let's work through the problem step-by-step.
1. Start with the given inequality:
[tex]\[
|x| - 4 \geq 10
\][/tex]
2. Isolate the absolute value:
[tex]\[
|x| \geq 10 + 4
\][/tex]
[tex]\[
|x| \geq 14
\][/tex]
3. Understand the meaning of absolute value:
The absolute value [tex]\( |x| \geq 14 \)[/tex] means that [tex]\( x \)[/tex] is either less than or equal to -14, or greater than or equal to 14. This gives us two separate inequalities:
[tex]\[
x \leq -14 \quad \text{or} \quad x \geq 14
\][/tex]
4. Combine the inequalities to form the solution set:
The solution set includes all [tex]\( x \)[/tex]-values that are either less than or equal to -14, or greater than or equal to 14. Written in interval notation, this is:
[tex]\[
x \leq -14 \quad \text{or} \quad x \geq 14
\][/tex]
In other words, [tex]\( x \)[/tex] is in the intervals [tex]\((-\infty, -14]\)[/tex] or [tex]\([14, \infty)\)[/tex].
5. Let's map these intervals to the options provided:
- Option a: [tex]\( x \geq 14 \)[/tex] (This only covers part of the solution set.)
- Option b: [tex]\( x \geq 14 \)[/tex] or [tex]\( x \leq -14 \)[/tex] (This matches exactly with our solution set.)
- Option c: [tex]\( x \leq -14 \)[/tex] or [tex]\( x \geq 6 \)[/tex] (This does not match, as [tex]\( x \geq 6 \)[/tex] is not correct.)
- Option d: [tex]\( -14 \leq x \leq 14 \)[/tex] (This covers values between -14 and 14 which is incorrect.)
Thus, the correct answer is:
[tex]\[
\boxed{\text{b. } x \geq 14 \text{ or } x \leq -14}
\][/tex]
