Let's solve the given problem step-by-step:
We have the expression [tex]\(3 \times (2x - 1)\)[/tex]. To find the product, we need to distribute the [tex]\(3\)[/tex] across the terms inside the parentheses. Here's the detailed step-by-step process:
1. Start with the given factors: [tex]\(3\)[/tex] and [tex]\((2x - 1)\)[/tex].
2. Distribute the [tex]\(3\)[/tex] to each term inside the parentheses:
[tex]\[
3 \times (2x - 1) = 3 \times 2x + 3 \times (-1)
\][/tex]
3. Multiply each term individually:
[tex]\[
3 \times 2x = 6x
\][/tex]
[tex]\[
3 \times (-1) = -3
\][/tex]
4. Combine the results:
[tex]\[
3 \times (2x - 1) = 6x - 3
\][/tex]
So, the product is [tex]\(6x - 3\)[/tex].
Now, let's consider the options regarding Armando's use of algebra tiles:
1. He used the algebra tiles correctly:
If Armando's algebra tiles represented [tex]\(3\)[/tex] groups of [tex]\(2x - 1\)[/tex] and combined them to show [tex]\(6x - 3\)[/tex], then he used the tiles correctly.
2. He did not represent the two original factors correctly:
If Armando's initial setup with algebra tiles did not correctly feature [tex]\(3\)[/tex] and [tex]\((2x - 1)\)[/tex] in appropriate quantities, then this statement would be true. However, based on the correct product of [tex]\(6x - 3\)[/tex], this seems unlikely.
3. The signs on some of the products are incorrect:
If Armando's tiles showed incorrect signs (for example, representing [tex]\(2x - 1\)[/tex] as [tex]\(2x + 1\)[/tex]), the final product wouldn't match [tex]\(6x - 3\)[/tex]. But since the correct product is given, this statement would be false.
4. Some of the products do not show the correct powers of [tex]\(x\)[/tex]:
The problem only involves the first power of [tex]\(x\)[/tex]; if Armando’s product was correct, then all powers (in this case, just [tex]\(x\)[/tex]) would be shown correctly.
Since the product [tex]\(6x - 3\)[/tex] matches our expanded expression exactly, we can conclude that Armando used the algebra tiles correctly. Hence, the true statement regarding Armando's use of algebra tiles is:
- He used the algebra tiles correctly.
