To solve the inequality [tex]\( |6x - 9| > 15 \)[/tex], we need to consider the definition of absolute value and break the inequality into two separate cases. The absolute value inequality [tex]\( |A| > B \)[/tex] can be split into two inequalities: [tex]\( A > B \)[/tex] and [tex]\( A < -B \)[/tex]. Let's apply this to our problem.
Given:
[tex]\[ |6x - 9| > 15 \][/tex]
We break it into:
1. [tex]\( 6x - 9 > 15 \)[/tex]
2. [tex]\( 6x - 9 < -15 \)[/tex]
Let's solve these inequalities one by one.
Case 1: [tex]\( 6x - 9 > 15 \)[/tex]
Add 9 to both sides:
[tex]\[ 6x > 24 \][/tex]
Divide both sides by 6:
[tex]\[ x > 4 \][/tex]
Case 2: [tex]\( 6x - 9 < -15 \)[/tex]
Add 9 to both sides:
[tex]\[ 6x < -6 \][/tex]
Divide both sides by 6:
[tex]\[ x < -1 \][/tex]
So, the solution to the inequality [tex]\( |6x - 9| > 15 \)[/tex] is the union of the intervals [tex]\( x < -1 \)[/tex] and [tex]\( x > 4 \)[/tex]. In interval notation, this can be expressed as:
[tex]\[ (-\infty, -1) \cup (4, \infty) \][/tex]
Therefore, the solution set is:
[tex]\[ \boxed{\text{Union(Interval.open(-oo, -1), Interval.open(4, oo))}} \][/tex]
This is the correct graph that shows the solution, where the regions outside the intervals [tex]\([-1, 4]\)[/tex] are shaded to indicate that these [tex]\( x \)[/tex]-values satisfy the inequality.
