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Answer
The solution to the inequality \( |x - 9| \geq 3 \) is all real numbers \( x \) such that \( x \leq 6 \) or \( x \geq 12 \).
Solution
To solve the inequality:
\[
|x - 9| \geq 3
\]
we need to consider the definition of absolute value. Recall that for any real number \( a \):
\[
|a| \geq b \quad \text{if and only if} \quad a \geq b \quad \text{or} \quad a \leq -b
\]
Applying this to the given inequality:
\[
|x - 9| \geq 3 \quad \Rightarrow \quad x - 9 \geq 3 \quad \text{or} \quad x - 9 \leq -3
\]
Let's solve each inequality separately.
1. **First Inequality:**
\[
x - 9 \geq 3
\]
Add 9 to both sides:
\[
x \geq 3 + 9 \\
x \geq 12
\]
2. **Second Inequality:**
\[
x - 9 \leq -3
\]
Add 9 to both sides:
\[
x \leq -3 + 9 \\
x \leq 6
\]
**Combined Solution:**
The solution to the inequality \( |x - 9| \geq 3 \) is all real numbers \( x \) such that:
\[
x \leq 6 \quad \text{or} \quad x \geq 12
\]
**In Interval Notation:**
\[
(-\infty, 6] \cup [12, \infty)
\]
**Graphical Representation:**
```
<----|=========|----->
6 12
```
Here, the shaded regions extend to the left of 6 and to the right of 12, indicating all real numbers less than or equal to 6 or greater than or equal to 12 satisfy the inequality.
**Final Answer:**
\[
x \leq 6 \quad \text{or} \quad x \geq 12
\]
**Or in Interval Notation:**
\[
(-\infty, 6] \cup [12, \infty)
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To solve the inequality \( |x-9| \geq 3 \), we start by considering the definition of absolute value. The inequality can be split into two separate cases: 1. \( x - 9 \geq 3 \) 2. \( x - 9 \leq -3 \) For the first case: \[ x - 9 \geq 3 \implies x \geq 12 \] For the second case: \[ x - 9 \leq -3 \implies x \leq 6 \] Thus, the solution to the inequality \( |x-9| \geq 3 \) is: \[ x \leq 6 \quad \text{or} \quad x \geq 12 \] In interval notation, the solution is: \[ (-\infty, 6] \cup [12, \infty) \]
