Let’s go through each part of the problem.

(i) Solve the inequality: $-3x < 12$ and find the minimum integral solution $x$ can take.

1. Begin with the inequality: $-3x < 12$

2. Divide both sides by $-3$, and remember to reverse the inequality sign: $x > -4$

3. The minimum integer $x$ can take is $-3$.

Answer: $x > -4$, and the minimum integral solution is $x = -3$.

(ii) Factorize $x^2 - 9x - 36$.

1. To factorize $x^2 - 9x - 36$, we need two numbers that multiply to $-36$ and add to $-9$.

   • These numbers are $-12$ and $3$.

2. Rewrite the expression: $x^2 - 9x - 36 = (x - 12)(x + 3)$

Answer: $(x - 12)(x + 3)$.

(iii) Solve the system of equations:

$2a - 3b = -3$ $a + 3b = 12$

1. Add the two equations to eliminate $b$: $(2a - 3b) + (a + 3b) = -3 + 12$ $3a = 9$ $a = 3$

2. Substitute $a = 3$ into the second equation to solve for $b$: $3 + 3b = 12$ $3b = 9$ $b = 3$

Answer: $a = 3$, $b = 3$.

(iv) Simplify:

$\frac{5}{5x + y} - \frac{2 - x}{5x + y}$

1. Combine the terms since they have a common denominator: $\frac{5 - (2 - x)}{5x + y}$

2. Simplify the numerator: $= \frac{5 - 2 + x}{5x + y}$ $= \frac{3 + x}{5x + y}$

Answer: $\frac{3 + x}{5x + y}$.

Would you like further clarification on any of these solutions?

Here are 5 related questions to explore:

1. How would the inequality change if $-3x \geq 12$ in part (i)?
2. Can you apply a different method to factorize $x^2 - 9x - 36$ in part (ii)?
3. How would you solve the system if one of the equations was non-linear?
4. What if the denominators in part (iv) were different? How would you simplify then?
5. How can you verify the solutions in part (iii) after solving?

Tip: For factorization, always look for pairs of numbers that satisfy both the addition and multiplication criteria for faster results.