To solve the inequality [tex]\(4|x - 9| \geq 16\)[/tex], we need to handle the absolute value expression. We approach it by considering the definition of absolute value and translating the given inequality into two separate inequalities that do not involve absolute values.
1. Rewrite the Inequality:
[tex]\[ 4|x - 9| \geq 16 \][/tex]
2. Isolate the Absolute Value:
Divide both sides of the inequality by 4:
[tex]\[ |x - 9| \geq 4 \][/tex]
3. Remove the Absolute Value:
The absolute value inequality [tex]\(|x - 9| \geq 4\)[/tex] implies two cases:
- [tex]\(x - 9 \geq 4\)[/tex] (x is sufficiently greater than 9)
- [tex]\(x - 9 \leq -4\)[/tex] (x is sufficiently less than 9)
4. Solve Each Case:
- For [tex]\(x - 9 \geq 4\)[/tex]:
[tex]\[ x - 9 \geq 4 \][/tex]
Add 9 to both sides:
[tex]\[ x \geq 13 \][/tex]
- For [tex]\(x - 9 \leq -4\)[/tex]:
[tex]\[ x - 9 \leq -4 \][/tex]
Add 9 to both sides:
[tex]\[ x \leq 5 \][/tex]
5. Combine the Solutions:
The solutions from the two cases are [tex]\(x \geq 13\)[/tex] and [tex]\(x \leq 5\)[/tex].
Note: The inequality requires [tex]\(4|x - 9| \geq 16\)[/tex], meaning [tex]\(x\)[/tex] must be at a distance of at least 4 units from 9. Therefore, the valid solutions are:
[tex]\[
x \leq 5 \quad \text{or} \quad x \geq 13
\][/tex]
6. Conclude the Solution:
The solution set for the inequality [tex]\(4|x - 9| \geq 16\)[/tex] is:
[tex]\[
x \leq 5 \quad \text{or} \quad x \geq 13
\][/tex]
Hence, the answer is:
[tex]\[
x \leq 5 \quad \text{or} \quad x \geq 13
\][/tex]
This means the values of [tex]\(x\)[/tex] are either less than or equal to 5 or greater than or equal to 13.
