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!