To determine which expressions are equivalent to [tex]\(12r - 5\)[/tex], we need to simplify each given expression and compare it to [tex]\(12r - 5\)[/tex].
### Expression A: [tex]\(6(2r + (-1)) + 1\)[/tex]
1. Simplify inside the parentheses first:
[tex]\[
2r + (-1) = 2r - 1
\][/tex]
2. Multiply by 6:
[tex]\[
6 \cdot (2r - 1) = 6 \cdot 2r - 6 \cdot 1 = 12r - 6
\][/tex]
3. Add 1:
[tex]\[
12r - 6 + 1 = 12r - 5
\][/tex]
### Expression B: [tex]\(-4(2 + (-3r)) + 3\)[/tex]
1. Simplify inside the parentheses first:
[tex]\[
2 + (-3r) = 2 - 3r
\][/tex]
2. Multiply by [tex]\(-4\)[/tex]:
[tex]\[
-4 \cdot (2 - 3r) = -4 \cdot 2 + 4 \cdot 3r = -8 + 12r
\][/tex]
3. Add 3:
[tex]\[
-8 + 12r + 3 = 12r - 5
\][/tex]
### Comparison with [tex]\(12r - 5\)[/tex]
Both expressions, A and B, simplify to [tex]\(12r - 5\)[/tex]. Thus, both are equivalent to [tex]\(12r - 5\)[/tex].
### Conclusion
The expressions that are equivalent to [tex]\(12r - 5\)[/tex] are:
- A) [tex]\(6(2r + (-1)) + 1\)[/tex]
- B) [tex]\(-4(2 + (-3r)) + 3\)[/tex]
So, the correct answers are:
- A) [tex]\(6(2r + (-1)) + 1\)[/tex]
- B) [tex]\(-4(2 + (-3r)) + 3\)[/tex]
