https://file.aiquickdraw.com/mathbot/file/1726247431880ezpiyl54.jpg

Let's go through each option to check which statements are true:

### Statement A: \(|a - b| > 0\)
- The absolute value \(|a - b|\) measures the distance between \(a\) and \(b\) on the number line.
- \(|a - b|\) is always greater than or equal to 0, but it is equal to 0 when \(a = b\).
- So, \(|a - b|\) is **greater than 0** only when \(a \neq b\), but not always.
- **Not always true**, but true when \(a \neq b\).

### Statement B: \((\sqrt{77.77})(\sqrt{77.77}) = 77.77\)
- This is the property of square roots: \((\sqrt{x})(\sqrt{x}) = x\).
- In this case, \(\sqrt{77.77} \cdot \sqrt{77.77} = 77.77\).
- This is **true**.

### Statement C: \(-a + b = b - a\)
- Simplifying both sides:
  \[
  -a + b = b - a
  \]
  Both expressions are equal because the commutative property of addition holds.
- This is **true**.

### Statement D: \(a - b - c = b - a - c\)
- Rewriting both sides:
  \[
  a - b - c \quad \text{versus} \quad b - a - c
  \]
  This simplifies to:
  \[
  a - b - c \neq b - a - c
  \]
  These are not equivalent unless \(a = b\), so this is **not always true**.

### Statement E: \(a(0) > | -b - c |\)
- \(a(0)\) means multiplying \(a\) by zero, so \(a(0) = 0\).
- The right side is the absolute value of \(-b - c\), which will always be non-negative.
- Since 0 is not greater than any positive number, this is **false**.

### Final Answers:
- The true statements are **B** and **C**.

Would you like more details or explanations on any of the steps? Here are 5 questions to expand on this:

1. Why does the absolute value of a difference, \(|a - b|\), always yield a non-negative result?
2. How can the commutative property help simplify equations like \(-a + b = b - a\)?
3. What are the conditions under which statement D would be true?
4. What properties of square roots are useful for verifying statement B?
5. Why does multiplying by zero result in a false inequality in statement E?

**Tip**: Always check for the basic properties of operations (like commutative, associative, or distributive) when simplifying expressions!

Select ALL the statements that are true:
A. |a - b| > 0
B. (√77.77)(√77.77) = 77.77
C. -a + b = b - a
D. a - b - c = b - a - c
E. a(0) > | -b - c |

This problem asks students to evaluate various mathematical statements involving absolute value, square roots, commutative properties, and algebraic inequalities. The correct true statements are B and C, exploring key Grade 7 algebra concepts.

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