Properties of Real Numbers Worksheet with algebraic expressions and their corresponding mathematical properties.

Problem Description:

The task involves identifying the algebraic property being used in each given equation or expression. The properties listed on the right side of the image are to be matched with the corresponding equations on the left.

Solution:

Let's analyze each equation step by step and identify the property being used.

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#### 1. \( (x + 1) + 4 = x + (1 + 4) \)
- This equation shows that the grouping of terms does not affect the result when adding numbers.
- Property: Associative Property of Addition
- Reasoning: The associative property states that for any numbers \( a \), \( b \), and \( c \), \( (a + b) + c = a + (b + c) \).

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#### 2. \( xy = yx \)
- This equation shows that the order of multiplication does not affect the result.
- Property: Commutative Property of Multiplication
- Reasoning: The commutative property states that for any numbers \( a \) and \( b \), \( ab = ba \).

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#### 3. \( 8 + 1 = 8 \)
- This equation is incorrect as written because \( 8 + 1 \neq 8 \). However, if we interpret it as an example of adding zero (assuming a typo), then:
- Property: Identity Property of Addition
- Reasoning: The identity property of addition states that adding zero to any number leaves the number unchanged (\( a + 0 = a \)).

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#### 4. \( 3 \times 8 = 8 \times 3 \)
- This equation shows that the order of multiplication does not affect the result.
- Property: Commutative Property of Multiplication
- Reasoning: The commutative property states that for any numbers \( a \) and \( b \), \( ab = ba \).

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#### 5. \( 9 + 0 = 9 \)
- This equation shows that adding zero to any number leaves the number unchanged.
- Property: Identity Property of Addition
- Reasoning: The identity property of addition states that \( a + 0 = a \).

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#### 6. \( 15 \times 0 = 0 \)
- This equation shows that multiplying any number by zero results in zero.
- Property: Property of Zero
- Reasoning: The property of zero in multiplication states that \( a \times 0 = 0 \).

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#### 7. \( 99 + 0 = 99 \)
- This equation shows that adding zero to any number leaves the number unchanged.
- Property: Identity Property of Addition
- Reasoning: The identity property of addition states that \( a + 0 = a \).

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#### 8. \( x + 11 = 11 + x \)
- This equation shows that the order of addition does not affect the result.
- Property: Commutative Property of Addition
- Reasoning: The commutative property states that for any numbers \( a \) and \( b \), \( a + b = b + a \).

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#### 9. \( [2 + (1 + 3)] = 2 + [(1 + 3)] \)
- This equation shows that the grouping of terms does not affect the result when adding numbers.
- Property: Associative Property of Addition
- Reasoning: The associative property states that for any numbers \( a \), \( b \), and \( c \), \( (a + b) + c = a + (b + c) \).

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#### 10. \( 3 \times 9 = 9 \times 3 \)
- This equation shows that the order of multiplication does not affect the result.
- Property: Commutative Property of Multiplication
- Reasoning: The commutative property states that for any numbers \( a \) and \( b \), \( ab = ba \).

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#### 11. \( 4 + (5 + 6) = (4 + 5) + 6 \)
- This equation shows that the grouping of terms does not affect the result when adding numbers.
- Property: Associative Property of Addition
- Reasoning: The associative property states that for any numbers \( a \), \( b \), and \( c \), \( (a + b) + c = a + (b + c) \).

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#### 12. \( 10(2) = 20 \)
- This equation shows the basic operation of multiplication.
- Property: None (This is a straightforward computation, not a property.)

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#### 13. \( 1 + 0 = 1 \)
- This equation shows that adding zero to any number leaves the number unchanged.
- Property: Identity Property of Addition
- Reasoning: The identity property of addition states that \( a + 0 = a \).

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#### 14. \( 2(0) = 0 \)
- This equation shows that multiplying any number by zero results in zero.
- Property: Property of Zero
- Reasoning: The property of zero in multiplication states that \( a \times 0 = 0 \).

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Final Answer:

Matching the equations with their respective properties:

1. Associative Property of Addition
2. Commutative Property of Multiplication
3. Identity Property of Addition
4. Commutative Property of Multiplication
5. Identity Property of Addition
6. Property of Zero
7. Identity Property of Addition
8. Commutative Property of Addition
9. Associative Property of Addition
10. Commutative Property of Multiplication
11. Associative Property of Addition
12. None
13. Identity Property of Addition
14. Property of Zero

\[
\boxed{
\begin{array}{ll}
1. & \text{Associative Property of Addition} \\
2. & \text{Commutative Property of Multiplication} \\
3. & \text{Identity Property of Addition} \\
4. & \text{Commutative Property of Multiplication} \\
5. & \text{Identity Property of Addition} \\
6. & \text{Property of Zero} \\
7. & \text{Identity Property of Addition} \\
8. & \text{Commutative Property of Addition} \\
9. & \text{Associative Property of Addition} \\
10. & \text{Commutative Property of Multiplication} \\
11. & \text{Associative Property of Addition} \\
12. & \text{None} \\
13. & \text{Identity Property of Addition} \\
14. & \text{Property of Zero} \\
\end{array}
}
\]