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Let's solve the given compound inequality step by step. The compound inequality is:

\[
2y - 1 \leq 7 \quad \text{or} \quad 4y - 3 > 9
\]

### Step 1: Solve each inequality separately.

#### First inequality: \(2y - 1 \leq 7\)

1. Add 1 to both sides:
   \[
   2y \leq 7 + 1
   \]
   \[
   2y \leq 8
   \]

2. Divide both sides by 2:
   \[
   y \leq \frac{8}{2}
   \]
   \[
   y \leq 4
   \]

#### Second inequality: \(4y - 3 > 9\)

1. Add 3 to both sides:
   \[
   4y > 9 + 3
   \]
   \[
   4y > 12
   \]

2. Divide both sides by 4:
   \[
   y > \frac{12}{4}
   \]
   \[
   y > 3
   \]

### Step 2: Combine the solution.

The solution to the compound inequality involves a union of the two results:

- From the first inequality, \( y \leq 4 \).
- From the second inequality, \( y > 3 \).

Since this is an "or" inequality, we are looking for the union of these two conditions, which results in the interval:

\[
y > 3 \quad \text{or} \quad y \leq 4
\]

This means the solution is all values of \( y \) greater than 3, and it extends up to and including 4. In interval notation, this is:

\[
(3, 4]
\]

### Final Answer:
The solution set in interval notation is \( (3, 4] \).

Would you like any more details or explanations on this? Here are a few related questions to expand on this topic:

1. How do you solve compound inequalities with "and" instead of "or"?
2. How can compound inequalities be represented on a number line?
3. What happens if there are no overlapping solutions in a compound inequality?
4. How does interval notation work for inequalities involving infinity?
5. Can you have an inequality with both "and" and "or" conditions?

**Tip:** When solving compound inequalities, always check whether the statement uses "and" or "or", as this will affect how the solution is combined.

Solve the following compound inequality: 2y - 1 ≤ 7 or 4y - 3 > 9. Write your answer in interval notation or state that there is no solution.

Learn how to solve the compound inequality 2y - 1 ≤ 7 or 4y - 3 > 9 with a step-by-step solution. This problem covers key algebraic concepts and inequality rules, providing a detailed explanation. Suitable for students in Grades 9-11.

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