Let’s solve the inequalities one by one.

### a) \(3x + 14 > -9\)

To solve this inequality, we first isolate \(x\):

1. Subtract 14 from both sides:
\[
3x > -9 - 14
\]
\[
3x > -23
\]

2. Divide both sides by 3:
\[
x > -\frac{23}{3}
\]
\[
x > -7\frac{2}{3}
\]

### b) \(2x - 3 \geq 5x + 14\)

To solve this inequality, we can rearrange it:

1. Subtract \(2x\) from both sides:
\[
-3 \geq 3x + 14
\]

2. Next, subtract 14 from both sides:
\[
-3 - 14 \geq 3x
\]
\[
-17 \geq 3x
\]

3. Finally, divide both sides by 3, remembering to reverse the inequality:
\[
-\frac{17}{3} \geq x
\]
\[
x \leq -\frac{17}{3}
\]

### c) \(7x - 4(x - 2) < 3x + 5\)

Let’s simplify and solve this inequality:

1. Distributing \(-4\):
\[
7x - 4x + 8 < 3x + 5
\]

2. Combine like terms:
\[
3x + 8 < 3x + 5
\]

3. Subtract \(3x\) from both sides:
\[
8 < 5
\]

This is a false statement. Therefore, there is no solution for this inequality.

### Summary of Solutions:

- a) \(x > -\frac{23}{3}\)
- b) \(x \leq -\frac{17}{3}\)
- c) No solution.