Let 𝐴 = {𝑥 ∈ ℝ:−4 ≤ 𝑥 < 2} and 𝐵 = {𝑥 ∈ ℝ:𝑥 ≥ −1}. Draw and use the Venn diagrams to find  a. 𝐴 ∩ 𝐵            b.  𝐴′ 

To solve the given problem, we will start by analyzing the sets \( A \) and \( B \), then use their definitions to draw Venn diagrams and determine the required intersections and complements.

### Step 1: Define Sets \( A \) and \( B \)

Given:

- \( A = \{x \in \mathbb{R} : -4 \leq x < 2\} \)  
  This means that \( A \) includes all real numbers from \(-4\) to \(2\), where \(-4\) is included, but \(2\) is not.

- \( B = \{x \in \mathbb{R} : x \geq -1\} \)  
  This means that \( B \) includes all real numbers from \(-1\) and above.

### Step 2: Find the Intersection \( A \cap B \)

The intersection \( A \cap B \) includes all values of \( x \) that are in both sets.

- The set \( A \) ranges from \(-4\) to \(2\), excluding \(2\).
- The set \( B \) includes all numbers greater than or equal to \(-1\).

To find \( A \cap B \), we look for the overlap between these two intervals:

\[
A \cap B = \{x \in \mathbb{R} : -1 \leq x < 2\}
\]

### Step 3: Find the Complement of \( A \), Denoted as \( A' \)

The complement of \( A \) (\( A' \)) consists of all real numbers not in \( A \).

- Since \( A = \{x \in \mathbb{R} : -4 \leq x < 2\} \), the complement is:

\[
A' = \{x \in \mathbb{R} : x < -4\} \cup \{x \in \mathbb{R} : x \geq 2\}
\]

This represents all numbers less than \(-4\) or greater than or equal to \(2\).

### Venn Diagram Illustration

To visualize this, let's outline what the Venn diagram would look like:

1. **Set \( A \)**: Covers the interval from \(-4\) to \(2\), with an open endpoint at \(2\).
2. **Set \( B \)**: Starts at \(-1\) and extends infinitely to the right.
3. **Intersection \( A \cap B \)**: The overlapping region from \(-1\) to \(2\), excluding \(2\).
4. **Complement \( A' \)**: All regions outside of \( A \).

### Graphical Representation

Although I cannot directly draw the diagram here, here's how you could sketch it:

1. Draw two overlapping regions representing \( A \) and \( B \).
2. Mark the interval of \( A \) from \(-4\) to \(2\) and shade lightly.
3. Mark the interval of \( B \) from \(-1\) and shade it differently.
4. The overlapping part (intersection) represents \( A \cap B \).
5. Shade the area outside \( A \) to represent \( A' \).

Would you like me to generate a visual diagram of this setup? 

### Related Questions:
1. What is the union of \( A \) and \( B \)?
2. How would you represent the complement of \( B \), \( B' \)?
3. How do you calculate the symmetric difference between two sets?
4. Can you determine the difference \( A - B \)?
5. How can Venn diagrams be used to visualize more complex set operations?

**Tip:** When working with inequalities and intervals, drawing a number line or a Venn diagram can greatly help in visualizing the relationships between sets!

Explore set theory with a focus on intersection and complement operations. Learn how to find intersections and complements of sets represented by intervals. Suitable for high school students and those interested in set theory fundamentals.

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