To solve the absolute value inequality [tex]\(\frac{|x + 9|}{4} > 2\)[/tex], we need to isolate the absolute value expression and then split it into two separate inequalities.
### Step-by-Step Solution
1. Remove the fraction by multiplying both sides by 4:
[tex]\[
\frac{|x + 9|}{4} > 2
\][/tex]
Multiply both sides by 4:
[tex]\[
|x + 9| > 8
\][/tex]
2. Set up the two possible inequalities that arise from the absolute value expression:
The absolute value inequality [tex]\( |A| > B \)[/tex] means that [tex]\( A > B \)[/tex] or [tex]\( A < -B \)[/tex]. Therefore, we have:
[tex]\[
x + 9 > 8 \quad \text{or} \quad x + 9 < -8
\][/tex]
3. Solve each inequality separately:
- For [tex]\( x + 9 > 8 \)[/tex]:
[tex]\[
x > 8 - 9
\][/tex]
[tex]\[
x > -1
\][/tex]
- For [tex]\( x + 9 < -8 \)[/tex]:
[tex]\[
x < -8 - 9
\][/tex]
[tex]\[
x < -17
\][/tex]
### Final Solution
Combining the solutions from both parts, we have:
[tex]\[
x > -1 \quad \text{or} \quad x < -17
\][/tex]
### Conclusion
The solution to the inequality [tex]\(\frac{|x + 9|}{4} > 2\)[/tex] is:
[tex]\[
x > -1 \quad \text{or} \quad x < -17
\][/tex]
These solutions indicate that [tex]\( x \)[/tex] can be any number greater than [tex]\(-1\)[/tex] or any number less than [tex]\(-17\)[/tex]. Remember, there is no overlap between these two ranges, as they are mutually exclusive.
