Which of the following is equivalent to [tex]6(2x-4) \ \textgreater \ 3 + 2x[/tex]?

A. [tex]12x + 2 \ \textgreater \ 3 + 2x[/tex]
B. [tex]12x - 10 \ \textgreater \ 3 + 2x[/tex]
C. [tex]12x - 24 \ \textgreater \ 3 + 2x[/tex]
D. [tex]12x - 4 \ \textgreater \ 3 + 2x[/tex]



Answer :

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]