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### 1. **Compare Quantity A** 
   We need to evaluate both expressions in **Quantity A** and compare them using \( > \), \( < \), or \( = \).

- **First Expression**: 
   \[
   2 \left( \frac{1}{3} \right)^3 = 2 \times \left( \frac{1}{27} \right) = \frac{2}{27}
   \]

- **Second Expression**:
   \[
   4(-7)^3 + (3 + (-7))^3 = 4(-343) + (-4)^3 = -1372 + (-64) = -1436
   \]

Now compare the results:
\[
\frac{2}{27} \approx 0.074 \quad \text{and} \quad -1436
\]

Clearly,
\[
\frac{2}{27} > -1436
\]

### 2. **Types of Equations** 
   For each type of equation, we provide examples and explain why they have the indicated number of solutions:

- **a) An equation with one solution**: 
   \[
   2x + 3 = 7
   \]
   - Solving for \(x\):
     \[
     2x = 4 \quad \Rightarrow \quad x = 2
     \]
   This equation has one solution because there is only one value of \(x\) that satisfies it.

- **b) An equation with infinite solutions**: 
   \[
   2(x - 1) = 2x - 2
   \]
   - Simplifying both sides:
     \[
     2x - 2 = 2x - 2
     \]
   The equation is true for all values of \(x\), so it has infinitely many solutions.

- **c) An equation with no solutions**: 
   \[
   2x + 3 = 2x + 5
   \]
   - Simplifying both sides:
     \[
     3 = 5
     \]
   This is a contradiction, so there are no solutions.

### 3. **Solve the Inequality**
   The inequality to solve is:
   \[
   8h - 2 < -10 \quad \text{or} \quad -h + 1 \leq -2
   \]
   
   - **First Inequality**:
     \[
     8h - 2 < -10
     \]
     Add 2 to both sides:
     \[
     8h < -8
     \]
     Divide by 8:
     \[
     h < -1
     \]

   - **Second Inequality**:
     \[
     -h + 1 \leq -2
     \]
     Subtract 1 from both sides:
     \[
     -h \leq -3
     \]
     Multiply both sides by -1 (remember to flip the inequality sign):
     \[
     h \geq 3
     \]

   So the solution is:
   \[
   h < -1 \quad \text{or} \quad h \geq 3
   \]

In interval notation, the solution is:
\[
(-\infty, -1) \cup [3, \infty)
\]

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Would you like more details on any part? Here are 5 related questions to consider:

1. How do you solve cubic expressions like \((-7)^3\) by hand?
2. What is the general approach to comparing fractional and large negative numbers?
3. Can equations with variables on both sides ever have no solutions?
4. How does the interval notation change when inequalities include strict vs. non-strict inequalities?
5. What is the significance of combining multiple inequalities in a solution?

**Tip:** When comparing numbers, it's always a good idea to convert fractions and large numbers into decimals for easy comparison.

1. Use your calculator in each quantity, then compare.
Quantity A: 2 (1/3)^3 vs 4(-7)^3 + (3 + (-7))^3.
2. Give an example of each type of equation.
a) An equation with one solution.
b) An equation with infinite solutions.
c) An equation with no solutions.
3. Solve the inequality: 8h - 2 < -10 or -h + 1 <= -2.

Compare quantities involving exponents, solve equations with one, infinite, or no solutions, and practice solving inequalities. Perfect for students in Grades 7-9, this guide walks through complex expressions and their solutions.

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