a. Choose the correct interval notation for the inequality notation
x > −3

[-3, ∞)

(-3, ∞)

(-∞ , -3)

(-∞ , 3)
None of these

b. Choose the correct interval notation for the inequality notation
−3 < x ≤ 3 or 7 ≤ x

(-3, 3] ⋃ [7, ∞)

(-3, 3) ⋃ (7, ∞)

[-3, 3) ⋃ [7, ∞)

[-3, 3] ⋃ (7, ∞)
None of these

c. Choose the correct interval notation for the inequality notation
x ≤ −3 or 7 ≤ x

(-∞ , -3] ⋃ (7, ∞)

(-∞ , -3) ⋃ (7, ∞)

(-∞ , -3] ⋃ [7, ∞)

(-∞ , 3) ⋃ (7, ∞)
None of these

Let's analyze and solve each part:

### a. Choose the correct interval notation for the inequality notation \( x > -3 \):

The inequality \( x > -3 \) means that \( x \) starts just above \(-3\) and goes to infinity. In interval notation, we express this as:

\[
(-3, \infty)
\]

- \( -3 \) is not included, so we use a parenthesis.
- Infinity \( \infty \) is always accompanied by a parenthesis.

**Answer for (a):** \((-3, \infty)\)

---

### b. Choose the correct interval notation for the inequality notation \( -3 < x \leq 3 \) or \( 7 \leq x \):

This represents two intervals:

1. \( -3 < x \leq 3 \): \( x \) is between \(-3\) and \(3\), but \( -3 \) is not included while \( 3 \) is included.
   - Interval: \( (-3, 3] \)
   
2. \( 7 \leq x \): \( x \) starts at 7 (included) and goes to infinity.
   - Interval: \( [7, \infty) \)

The union (\(\cup\)) of these intervals is:
\[
(-3, 3] \cup [7, \infty)
\]

**Answer for (b):** \((-3, 3] \cup [7, \infty)\)

---

### c. Choose the correct interval notation for the inequality notation \( x \leq -3 \) or \( 7 \leq x \):

This represents two intervals:

1. \( x \leq -3 \): \( x \) starts from negative infinity up to \(-3\) (including \(-3\)).
   - Interval: \( (-\infty, -3] \)
   
2. \( 7 \leq x \): \( x \) starts at 7 (included) and goes to infinity.
   - Interval: \( [7, \infty) \)

The union (\(\cup\)) of these intervals is:
\[
(-\infty, -3] \cup [7, \infty)
\]

**Answer for (c):** \((- \infty, -3] \cup [7, \infty)\)

---

### Summary of correct answers:
- **a:** \( (-3, \infty) \)
- **b:** \( (-3, 3] \cup [7, \infty) \)
- **c:** \( (-\infty, -3] \cup [7, \infty) \)

Would you like further details or explanations on any of these solutions?

---

### 5 Related Questions:
1. How do you express strict inequalities in interval notation?
2. What is the difference between open and closed intervals in inequalities?
3. How would you express \(x \geq 2\) in interval notation?
4. How do you represent a union of intervals in set notation?
5. What does an intersection of two intervals mean in terms of inequalities?

### Tip:
Always use parentheses for strict inequalities (\(<\) or \(>\)) and brackets for inclusive inequalities (\(\leq\) or \(\geq\)).

a. Choose the correct interval notation for the inequality notation x > −3: [-3, ∞), (-3, ∞), (-∞ , -3), (-∞ , 3), None of these.
b. Choose the correct interval notation for the inequality notation −3 < x ≤ 3 or 7 ≤ x: (-3, 3] ⋃ [7, ∞), (-3, 3) ⋃ (7, ∞), [-3, 3) ⋃ [7, ∞), [-3, 3] ⋃ (7, ∞), None of these.
c. Choose the correct interval notation for the inequality notation x ≤ −3 or 7 ≤ x: (-∞ , -3] ⋃ (7, ∞), (-∞ , -3) ⋃ (7, ∞), (-∞ , -3] ⋃ [7, ∞), (-∞ , 3) ⋃ (7, ∞), None of these.

This problem explores how to choose the correct interval notation for inequalities such as x > −3, −3 < x ≤ 3 or 7 ≤ x, and x ≤ −3 or 7 ≤ x. Learn how to interpret inequality symbols and apply them in set notation, including unions and intervals. Suitable for students in Grades 9-12.

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