To determine which of the given expressions is equivalent to the inequality [tex]\( 4(3x - 1) < -2(3x + 2) \)[/tex], we can follow these steps:
1. Distribute the numbers outside the parentheses on both sides of the inequality:
For the left side:
[tex]\[
4(3x - 1) = 4 \cdot 3x - 4 \cdot 1 = 12x - 4
\][/tex]
For the right side:
[tex]\[
-2(3x + 2) = -2 \cdot 3x - 2 \cdot 2 = -6x - 4
\][/tex]
2. Combine the results from both sides into one inequality:
[tex]\[
12x - 4 < -6x - 4
\][/tex]
3. Look at the choices provided to determine which one matches this inequality:
We are given four options:
[tex]\[
\begin{aligned}
&1) \quad 12x - 1 < 6x + 2 \\
&2) \quad 12x - 4 < -6x - 4 \\
&3) \quad 12x - 5 < -6x \\
&4) \quad 12x - 4 < 6x - 4 \\
\end{aligned}
\][/tex]
4. Compare each option:
- The first option, [tex]\( 12x - 1 < 6x + 2 \)[/tex], is not equivalent.
- The second option, [tex]\( 12x - 4 < -6x - 4 \)[/tex], matches exactly with the inequality we derived.
- The third option, [tex]\( 12x - 5 < -6x \)[/tex], is not equivalent.
- The fourth option, [tex]\( 12x - 4 < 6x - 4 \)[/tex], is not equivalent.
Thus, the correct equivalence is:
[tex]\[
12x - 4 < -6x - 4
\][/tex]
So, the answer to the question is [tex]\( \boxed{12x - 4 < -6x - 4} \)[/tex].
