Answer :
Let's analyze Armando's algebra tiles representation step-by-step to determine which statements about his work are true based on the given expression \(3 \times (2x - 1)\).
### Step-by-Step Solution:
1. Understanding the Expression:
- The expression Armando is working with is \(3 \times (2x - 1)\).
- This means he needs to distribute the 3 across the terms inside the parentheses:
[tex]\[ 3 \times (2x - 1) = 3 \times 2x + 3 \times (-1) \][/tex]
2. Distributing the 3:
- First, we multiply 3 by \(2x\):
[tex]\[ 3 \times 2x = 6x \][/tex]
- Second, we multiply 3 by \(-1\):
[tex]\[ 3 \times (-1) = -3 \][/tex]
3. Combining the Results:
- The simplified expression after distribution is:
[tex]\[ 6x - 3 \][/tex]
4. Analyzing Armando's Algebra Tiles:
- According to the distribution:
- Armando should have tiles representing \(6x\): six \(x\)-tiles.
- He should also have tiles representing \(-3\): three negative 1-tiles.
5. Verifying Armando's Representation:
- Armando's representation would be correct if:
- He used 6 \(x\)-tiles to represent \(6x\).
- He used three negative 1-tiles to represent \(-3\).
- He should have used the tiles correctly, representing each factor as \(6x - 3\).
- The signs of the products (positive and negative) should be accurate.
- All terms should reflect the correct powers of \(x\), which in this case, is simply \(x\).
### Conclusion:
Based on the analysis, we determine the following:
- Armando used the algebra tiles correctly.
- He represented the original factors (3 and \(2x - 1\)) correctly.
- The signs on the products (\(6x\) and \(-3\)) are correct.
- The powers of \(x\) are represented correctly (as just \(x\)).
Thus, the only true statement is:
- He used the algebra tiles correctly.
Therefore, the correct answer is:
- He used the algebra tiles correctly.
### Step-by-Step Solution:
1. Understanding the Expression:
- The expression Armando is working with is \(3 \times (2x - 1)\).
- This means he needs to distribute the 3 across the terms inside the parentheses:
[tex]\[ 3 \times (2x - 1) = 3 \times 2x + 3 \times (-1) \][/tex]
2. Distributing the 3:
- First, we multiply 3 by \(2x\):
[tex]\[ 3 \times 2x = 6x \][/tex]
- Second, we multiply 3 by \(-1\):
[tex]\[ 3 \times (-1) = -3 \][/tex]
3. Combining the Results:
- The simplified expression after distribution is:
[tex]\[ 6x - 3 \][/tex]
4. Analyzing Armando's Algebra Tiles:
- According to the distribution:
- Armando should have tiles representing \(6x\): six \(x\)-tiles.
- He should also have tiles representing \(-3\): three negative 1-tiles.
5. Verifying Armando's Representation:
- Armando's representation would be correct if:
- He used 6 \(x\)-tiles to represent \(6x\).
- He used three negative 1-tiles to represent \(-3\).
- He should have used the tiles correctly, representing each factor as \(6x - 3\).
- The signs of the products (positive and negative) should be accurate.
- All terms should reflect the correct powers of \(x\), which in this case, is simply \(x\).
### Conclusion:
Based on the analysis, we determine the following:
- Armando used the algebra tiles correctly.
- He represented the original factors (3 and \(2x - 1\)) correctly.
- The signs on the products (\(6x\) and \(-3\)) are correct.
- The powers of \(x\) are represented correctly (as just \(x\)).
Thus, the only true statement is:
- He used the algebra tiles correctly.
Therefore, the correct answer is:
- He used the algebra tiles correctly.
