Which of the following is equivalent to [tex]$4(3x - 1) \ \textless \ -2(3x + 2)$[/tex]?

A. [tex]$12x - 1 \ \textless \ 6x + 2$[/tex]
B. [tex][tex]$12x - 4 \ \textless \ -6x - 4$[/tex][/tex]
C. [tex]$12x - 5 \ \textless \ -6x$[/tex]
D. [tex]$12x - 4 \ \textless \ 6x - 4$[/tex]



Answer :

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].

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