To solve the absolute value inequality [tex]\( |x| > 6 \)[/tex], we need to understand what the absolute value function represents and how to handle inequalities involving it.
The absolute value [tex]\( |x| \)[/tex] of [tex]\( x \)[/tex] represents the distance of [tex]\( x \)[/tex] from 0 on the number line, regardless of the direction. When solving [tex]\( |x| > 6 \)[/tex], we are looking for the values of [tex]\( x \)[/tex] whose distance from 0 is greater than 6.
The inequality [tex]\( |x| > 6 \)[/tex] can be split into two separate inequalities:
1. [tex]\( x > 6 \)[/tex]
2. [tex]\( x < -6 \)[/tex]
This is because if [tex]\( x \)[/tex] is greater than 6 or less than -6, the distance from 0 will be more than 6 units in either direction.
So the complete solution for the inequality [tex]\( |x| > 6 \)[/tex] is the union of the two regions defined by [tex]\( x > 6 \)[/tex] and [tex]\( x < -6 \)[/tex].
Therefore, among the given answer choices:
a. [tex]\( -6 < x < 6 \)[/tex] is incorrect because it represents values of [tex]\( x \)[/tex] whose distance from 0 is less than 6.
b. [tex]\( x > 6 \)[/tex] or [tex]\( x < -6 \)[/tex] is correct as it accurately reflects the values of [tex]\( x \)[/tex] whose distance from 0 is greater than 6.
c. [tex]\( x = 6 \)[/tex] is incorrect because [tex]\( x \)[/tex] must be strictly greater than 6 or less than -6.
d. [tex]\( 0 < x < 6 \)[/tex] is incorrect because it represents values of [tex]\( x \)[/tex] whose distance from 0 is less than 6.
Thus, the correct choice is:
b. [tex]\( x > 6 \)[/tex] or [tex]\( x < -6 \)[/tex]
