find the truth set Predicate: 
1 ≤ x^2 ≤ 9,
 domain: R 

We are tasked with finding the truth set of the predicate \( 1 \leq x^2 \leq 9 \) with the domain \( x \in \mathbb{R} \) (the set of real numbers). The truth set refers to the set of all values of \( x \) that satisfy the given predicate.

### Step 1: Break down the inequality
The given predicate is a double inequality:

\[
1 \leq x^2 \leq 9
\]

This means that \( x^2 \) must satisfy both of the following conditions:
- \( x^2 \geq 1 \)
- \( x^2 \leq 9 \)

### Step 2: Solve the inequalities

1. **Solve \( x^2 \geq 1 \):**
   - \( x^2 = 1 \) when \( x = \pm 1 \).
   - For \( x^2 \geq 1 \), this inequality is satisfied when \( x \leq -1 \) or \( x \geq 1 \). Hence, \( x \in (-\infty, -1] \cup [1, \infty) \).

2. **Solve \( x^2 \leq 9 \):**
   - \( x^2 = 9 \) when \( x = \pm 3 \).
   - For \( x^2 \leq 9 \), this inequality is satisfied when \( -3 \leq x \leq 3 \). Hence, \( x \in [-3, 3] \).

### Step 3: Find the intersection
The truth set must satisfy both inequalities simultaneously, which means we need to find the intersection of the two solution sets:

- From \( x^2 \geq 1 \), we have \( x \in (-\infty, -1] \cup [1, \infty) \).
- From \( x^2 \leq 9 \), we have \( x \in [-3, 3] \).

The intersection of these two sets is:

\[
x \in [-3, -1] \cup [1, 3]
\]

### Final Answer:
The truth set of the predicate \( 1 \leq x^2 \leq 9 \) with \( x \in \mathbb{R} \) is:

\[
\boxed{[-3, -1] \cup [1, 3]}
\]

Would you like a more detailed explanation or have any other questions?

### Related questions:
1. How would the solution change if the inequality were \( 1 < x^2 < 9 \)?
2. What would be the truth set if the domain were restricted to integers?
3. How do you find the truth set of \( x^2 \geq 4 \) on the real number line?
4. Can you solve \( 2 \leq x^2 + x \leq 10 \) for \( x \in \mathbb{R} \)?
5. What is the graphical representation of the solution \( [-3, -1] \cup [1, 3] \)?

### Tip:
When solving compound inequalities, it's helpful to break them down into separate inequalities and then find the intersection of their solution sets.

Find the truth set Predicate: 1 ≤ x^2 ≤ 9, domain: R

Explore how to find the truth set of the inequality 1 ≤ x^2 ≤ 9 in the domain of real numbers. This detailed solution walks through the steps of solving quadratic inequalities and finding the intersection of solution sets. Ideal for high school students in Grades 9-12.

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