Let’s go through each inequality and solve them step by step:

a) $2x + 6 > -x + 9$

1. First, add $x$ to both sides to get all terms with $x$ on one side: $2x + x + 6 > 9$ $3x + 6 > 9$

2. Now subtract 6 from both sides: $3x > 3$

3. Finally, divide by 3: $x > 1$

b) $4x + 20 < 2x + 14$

1. Subtract $2x$ from both sides to isolate the variable $x$: $4x - 2x + 20 < 14$ $2x + 20 < 14$

2. Subtract 20 from both sides: $2x < -6$

3. Divide by 2: $x < -3$

c) $-x + 5 \geq 4x + 25$

1. Add $x$ to both sides to get rid of the negative $x$ on the left: $5 \geq 5x + 25$

2. Subtract 25 from both sides: $-20 \geq 5x$

3. Divide by 5: $-4 \geq x \quad \text{or} \quad x \leq -4$

d) $9P + 3 \leq -6$

1. Subtract 3 from both sides: $9P \leq -9$

2. Divide by 9: $P \leq -1$

Solutions:

a) $x > 1$
b) $x < -3$
c) $x \leq -4$
d) $P \leq -1$

Would you like more details on any of these steps?

Here are five related questions to consider:

1. How can inequalities change when multiplying or dividing by negative numbers?
2. What happens if we swap the sides of an inequality? Does the direction of the inequality change?
3. Can you explain the difference between "greater than" and "greater than or equal to"?
4. How would these inequalities be represented on a number line?
5. How would you handle inequalities with fractions?

Tip: When multiplying or dividing both sides of an inequality by a negative number, always remember to reverse the inequality symbol!