https://file.aiquickdraw.com/mathbot/file/17280495144792pw3rqz4.jpg

Let's go through the questions one by one and solve them.

### Question 4
**Given:**  
\[
\frac{2x + y}{x + 4y} = \frac{1}{2}
\]

#### (a) Express \( x \) in terms of \( y \):

We start by cross-multiplying to eliminate the fraction:

\[
2(2x + y) = 1(x + 4y)
\]

Expanding both sides:

\[
4x + 2y = x + 4y
\]

Now, collect like terms:

\[
4x - x = 4y - 2y
\]

Simplifying:

\[
3x = 2y
\]

So:

\[
x = \frac{2y}{3}
\]

---

#### (b) If \( y \) and \( 25 - xy \) are both positive integers, how many possible values of \( y \) are there?

From part (a), we know \( x = \frac{2y}{3} \). Substituting this into the condition \( 25 - xy \):

\[
25 - x \cdot y = 25 - \left(\frac{2y}{3}\right)y = 25 - \frac{2y^2}{3}
\]

For this expression to be a positive integer, \( y \) must be chosen so that \( \frac{2y^2}{3} \) is an integer. This implies that \( y^2 \) must be divisible by 3, meaning \( y \) must be a multiple of 3. Now, let's check values of \( y \) such that \( 25 - \frac{2y^2}{3} > 0 \).

---

### Question 5
**Given:**  
Solve the inequality \(\frac{22x - 17}{3} \leq 4x \).

#### (a) Solve for \( x \):

Start by multiplying both sides by 3 to eliminate the denominator:

\[
22x - 17 \leq 12x
\]

Now, subtract \( 12x \) from both sides:

\[
10x - 17 \leq 0
\]

Add 17 to both sides:

\[
10x \leq 17
\]

Divide by 10:

\[
x \leq \frac{17}{10} \quad \text{or} \quad x \leq 1.7
\]

---

#### (b) Greatest integer satisfying the inequality:

The greatest integer \( x \) such that \( x \leq 1.7 \) is \( x = 1 \).

---

### Question 6
**Given:**  
Solve the inequality \( \frac{9x}{2} > 3x + 5 \).

#### (a) Solve for \( x \):

Multiply both sides by 2 to eliminate the denominator:

\[
9x > 6x + 10
\]

Subtract \( 6x \) from both sides:

\[
3x > 10
\]

Divide by 3:

\[
x > \frac{10}{3} \quad \text{or} \quad x > 3.33
\]

---

#### (b) Is there a negative number that satisfies the inequality?

No, because the solution \( x > 3.33 \) indicates that only positive numbers greater than \( 3.33 \) satisfy the inequality.

---

### Question 7
If \( p \) and \( q \) are constants such that:
\[
x^2 - (3 + q)x + q \equiv (x - p)(x + 4) + 33
\]
find the values of \( p \) and \( q \).

We expand the right-hand side:

\[
(x - p)(x + 4) = x^2 + 4x - px - 4p = x^2 + (4 - p)x - 4p
\]

Now add 33:

\[
x^2 + (4 - p)x - 4p + 33
\]

Compare this with the left-hand side:

\[
x^2 - (3 + q)x + q
\]

From the comparison of coefficients, we get:

- \( 4 - p = -(3 + q) \), so \( p + q = 7 \)
- \( -4p + 33 = q \)

Now solve this system of equations.

---

### Question 8
Solve \( a + 2b = 4a + 3b = 5 \).

This can be rewritten as a system of two equations:

\[
a + 2b = 5 \quad \text{and} \quad 4a + 3b = 5
\]

Now solve this system.

---

Would you like detailed solutions for any of these parts or have any other questions? Here are some related questions to explore:

1. How do we find all integer solutions for inequalities?
2. What strategies can be used to solve a system of nonlinear equations?
3. What happens if the constants in Question 7 change?
4. How do we approach solving simultaneous equations algebraically?
5. What is the significance of boundary points in inequality solutions?

**Tip:** When solving inequalities, always consider the possibility of both integer and non-integer solutions, especially when asked to find the largest or smallest integer that satisfies the condition.

4. It is given that (2x + y) / (x + 4y) = 1/2.
(a) Express x in terms of y.
(b) If y and 25 - xy are both positive integers, how many possible values of y are there?
5. Solve the inequality 22x - 17 / 3 ≤ 4x.
(a) Solve the inequality.
(b) Write down the greatest integer satisfying the inequality in (a).
6. Solve the inequality 9x/2 > 3x + 5.
(a) Solve for x.
(b) Is there a negative number satisfying the inequality in (a)? Explain your answer.
7. If p and q are constants such that x^2 - (3 + q)x + q ≡ (x - p)(x + 4) + 33, find the values of p and q.
8. Solve a + 2b = 4a + 3b = 5.

This set of problems explores solving linear and quadratic inequalities, cross-multiplication methods, and systems of simultaneous equations. Learn how to solve equations like (2x + y) / (x + 4y) = 1/2, and the inequality 9x/2 > 3x + 5, with step-by-step solutions suitable for high school students in Grades 10-12.

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