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### 1. \( 2x + 9 \geq 3 \)
- Subtract 9 from both sides:
  \[
  2x \geq 3 - 9
  \]
  \[
  2x \geq -6
  \]
- Divide by 2:
  \[
  x \geq -3
  \]

### 2. \( 5x + 2 > 3x + 8 \)
- Subtract \(3x\) from both sides:
  \[
  2x + 2 > 8
  \]
- Subtract 2 from both sides:
  \[
  2x > 6
  \]
- Divide by 2:
  \[
  x > 3
  \]

### 3. \( 9x - 4 \leq 5x - 8 \)
- Subtract \(5x\) from both sides:
  \[
  4x - 4 \leq -8
  \]
- Add 4 to both sides:
  \[
  4x \leq -4
  \]
- Divide by 4:
  \[
  x \leq -1
  \]

### 4. \( 5x - 4 > 6 \)
- Add 4 to both sides:
  \[
  5x > 10
  \]
- Divide by 5:
  \[
  x > 2
  \]

### 5. \( 7x - 5 < 3x - 11 \)
- Subtract \(3x\) from both sides:
  \[
  4x - 5 < -11
  \]
- Add 5 to both sides:
  \[
  4x < -6
  \]
- Divide by 4:
  \[
  x < -\frac{3}{2}
  \]

### 6. \( 2x - 7 \leq 5x - 2 \)
- Subtract \(2x\) from both sides:
  \[
  -7 \leq 3x - 2
  \]
- Add 2 to both sides:
  \[
  -5 \leq 3x
  \]
- Divide by 3:
  \[
  x \geq -\frac{5}{3}
  \]

### 7. \( x + 8 < 6x - 5 \)
- Subtract \(x\) from both sides:
  \[
  8 < 5x - 5
  \]
- Add 5 to both sides:
  \[
  13 < 5x
  \]
- Divide by 5:
  \[
  x > \frac{13}{5}
  \]

### 8. \( 2x + 3 > x - 1 \)
- Subtract \(x\) from both sides:
  \[
  x + 3 > -1
  \]
- Subtract 3 from both sides:
  \[
  x > -4
  \]

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Would you like further details or explanations? Here are some follow-up questions:

1. How do you graph the solutions of these inequalities on a number line?
2. How do you handle inequalities when multiplying or dividing by a negative number?
3. What happens when two inequalities are combined using "and" or "or"?
4. How can you express the solution of an inequality in interval notation?
5. How can you verify the solutions by substituting values into the original inequalities?

**Tip:** Always reverse the inequality symbol when multiplying or dividing both sides by a negative number!

This page explains how to solve a variety of linear inequalities. It provides a step-by-step approach to solving each inequality, demonstrating the core algebraic principles involved in manipulating inequalities. Topics covered include properties of inequalities and how to express solutions.

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