To determine which of the provided inequalities is equivalent to the inequality [tex]\(6(2x - 4) > 3 + 2x\)[/tex], we'll go through a step-by-step process.
1. Distribute the 6 on the left-hand side of the inequality:
[tex]\[
6(2x - 4) > 3 + 2x
\][/tex]
This involves multiplying 6 with each term inside the parenthesis:
[tex]\[
6 \cdot 2x - 6 \cdot 4 > 3 + 2x
\][/tex]
Simplifying this, we get:
[tex]\[
12x - 24 > 3 + 2x
\][/tex]
2. Compare the simplified inequality with the given options:
- [tex]\(12x + 2 > 3 + 2x\)[/tex]
- [tex]\(12x - 10 > 3 + 2x\)[/tex]
- [tex]\(12x - 24 > 3 + 2x\)[/tex]
- [tex]\(12x - 4 > 3 + 2x\)[/tex]
3. Check which of these options matches the simplified inequality [tex]\(12x - 24 > 3 + 2x\)[/tex]:
From the list of options, the third choice:
[tex]\[
12x - 24 > 3 + 2x
\][/tex]
is exactly equivalent to our simplified inequality.
Therefore, the right choice is:
[tex]\[
12x - 24 > 3 + 2x
\][/tex]
Thus, the equivalent inequality is:
[tex]\[
\boxed{12x - 24 > 3 + 2x}
\][/tex]
