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Let's solve the compound inequality step by step.

We are given two inequalities:

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

We will solve each inequality separately, and since it is an "or" compound inequality, the solution will be the union of the two sets.

### Solving the first inequality:
\[
4x - 5 \leq 3
\]
Add 5 to both sides:
\[
4x \leq 8
\]
Now, divide by 4:
\[
x \leq 2
\]

### Solving the second inequality:
\[
2x + 5 \geq 3
\]
Subtract 5 from both sides:
\[
2x \geq -2
\]
Now, divide by 2:
\[
x \geq -1
\]

### Final solution:
Since this is an "or" inequality, the solution is \( x \leq 2 \) or \( x \geq -1 \). This means any \( x \) less than or equal to 2 or any \( x \) greater than or equal to -1 satisfies the inequality. The union of these two intervals is \( (-\infty, 2] \cup [-1, \infty) \), which simplifies to \( (-\infty, \infty) \).

Thus, the solution is the entire real line.

### Answer in interval notation:
\[
(-\infty, \infty)
\]

If you have any questions or would like further explanation, feel free to ask!

Here are 5 related questions to explore:
1. How do you solve a compound inequality that involves "and" instead of "or"?
2. How would the solution change if the inequalities involved strict inequalities (i.e., \( < \) or \( > \))?
3. Can compound inequalities have no solution? If so, how would that happen?
4. What are the differences between conjunction and disjunction in inequalities?
5. How can you graph the solution to a compound inequality?

**Tip:** When solving compound inequalities, always pay attention to whether the inequalities are connected by "and" or "or" as this greatly affects the solution.

Solve the compound inequality: 4x - 5 ≤ 3 or 2x + 5 ≥ 3. Write the solution in interval notation.

Learn how to solve the compound inequality 4x - 5 ≤ 3 or 2x + 5 ≥ 3. This guide covers essential algebraic steps, solving inequalities, and writing the solution in interval notation. Perfect for students in Grades 8-10.

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