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Let us solve the given inequalities step by step.

---

### **(i)**
We are solving the inequalities:
\[
1 + 4x \leq 9 \quad \text{and} \quad -\frac{2}{5} \leq x \leq 2.
\]

#### Step 1: Solve \( 1 + 4x \leq 9 \)
Subtract 1 from both sides:
\[
4x \leq 8.
\]
Divide both sides by 4:
\[
x \leq 2.
\]

#### Step 2: Combine with \( -\frac{2}{5} \leq x \leq 2 \)
The overlapping range of \( -\frac{2}{5} \leq x \leq 2 \) and \( x \leq 2 \) is:
\[
-\frac{2}{5} \leq x \leq 2.
\]

**Solution for (i):**
\[
-\frac{2}{5} \leq x \leq 2.
\]

---

### **(ii)**
We are solving the inequalities:
\[
3(x + 1) \geq -6 \quad \text{and} \quad \frac{3x}{2} + 1 < 7.
\]

#### Step 1: Solve \( 3(x + 1) \geq -6 \)
Expand the left-hand side:
\[
3x + 3 \geq -6.
\]
Subtract 3 from both sides:
\[
3x \geq -9.
\]
Divide both sides by 3:
\[
x \geq -3.
\]

#### Step 2: Solve \( \frac{3x}{2} + 1 < 7 \)
Subtract 1 from both sides:
\[
\frac{3x}{2} < 6.
\]
Multiply both sides by \(\frac{2}{3}\):
\[
x < 4.
\]

#### Step 3: Combine \( x \geq -3 \) and \( x < 4 \)
The overlapping range is:
\[
-3 \leq x < 4.
\]

**Solution for (ii):**
\[
-3 \leq x < 4.
\]

---

### Final Answer:
- (i): \( -\frac{2}{5} \leq x \leq 2 \)
- (ii): \( -3 \leq x < 4 \)

Would you like a detailed explanation for any part of this solution?

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### Related Questions:
1. How do you graph the solution of simultaneous linear inequalities?
2. What is the difference between solving an inequality and an equation?
3. How can simultaneous inequalities be applied in real-life scenarios?
4. What methods are used to check solutions for inequalities?
5. How do inequalities change when multiplying or dividing by a negative number?

**Tip:** Always verify overlapping regions of solutions carefully when solving simultaneous inequalities!

Solve each of the following simultaneous linear inequalities and find the possible values of x:
(i) 1 + 4x ≤ 9 and -2/5 ≤ x ≤ 2
(ii) 3(x + 1) ≥ -6 and (3x/2) + 1 < 7

Learn how to solve simultaneous linear inequalities, such as (i) 1 + 4x ≤ 9 and -2/5 ≤ x ≤ 2, and (ii) 3(x + 1) ≥ -6 and (3x/2) + 1 < 7. This detailed solution includes combining overlapping regions and applying linear inequality rules.

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