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Let's break down the problem and solve the inequalities step by step.

### Problem:
You are given two inequalities connected by the logical "AND":

1. \( 2x + 1 > 4x - 3 \)
2. \( x - 1 \geq 3x + 5 \)

We need to:
- Solve these inequalities.
- Express the solution set in interval and set-builder notation.
- Graph the solution on a number line.

### 1. Solve the first inequality: \( 2x + 1 > 4x - 3 \)

Subtract \( 2x \) from both sides:
\[
1 > 2x - 3
\]
Add 3 to both sides:
\[
4 > 2x
\]
Divide both sides by 2:
\[
2 > x
\]
or equivalently:
\[
x < 2
\]

### 2. Solve the second inequality: \( x - 1 \geq 3x + 5 \)

Subtract \( x \) from both sides:
\[
-1 \geq 2x + 5
\]
Subtract 5 from both sides:
\[
-6 \geq 2x
\]
Divide both sides by 2:
\[
-3 \geq x
\]
or equivalently:
\[
x \leq -3
\]

### Combine the solution sets

Since the two inequalities are connected by "AND," the solution set must satisfy both:
- \( x < 2 \)
- \( x \leq -3 \)

Thus, the solution set is \( x \leq -3 \), since this is the only region where both inequalities hold.

### Interval Notation:
The solution in interval notation is:
\[
(-\infty, -3]
\]

### Set-Builder Notation:
The solution in set-builder notation is:
\[
\{ x \mid x \leq -3 \}
\]

### Graph the Solution on a Number Line:
To graph the solution \( (-\infty, -3] \):
- Draw a number line.
- Mark the point \( -3 \) with a filled circle (indicating that \( x = -3 \) is included).
- Shade everything to the left of \( -3 \), towards negative infinity.

Would you like me to provide any further details, or do you have any questions?

---

Here are 5 related questions to expand your understanding:
1. How would the solution change if the inequalities were connected by "OR" instead of "AND"?
2. What would happen if the inequalities had different coefficients for \( x \)?
3. How can you graph the union of two inequalities on the number line?
4. What is the difference between solving inequalities with "AND" and "OR"?
5. How can you verify the solution to an inequality algebraically?

**Tip**: Always check the boundaries of your solution (like \( x = -3 \) here) to ensure it satisfies the original inequality before concluding the solution.

Solve and graph the solution set on a number line. Express the solution set in interval notation and set-builder notation: 2x + 1 > 4x - 3 AND x - 1 ≥ 3x + 5.

Learn how to solve and graph the compound inequalities 2x + 1 > 4x - 3 and x - 1 ≥ 3x + 5 on a number line. This problem explores algebraic methods for solving inequalities and expresses the solution in interval and set-builder notation. Suitable for students in Grades 8-10.

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