To solve the inequality [tex]\(-2(3x + 6) \geq 4(x + 7)\)[/tex], follow these steps:
1. Expand the expressions on both sides:
- Distribute [tex]\(-2\)[/tex] on the left-hand side:
[tex]\[
-2(3x + 6) = -6x - 12
\][/tex]
- Distribute [tex]\(4\)[/tex] on the right-hand side:
[tex]\[
4(x + 7) = 4x + 28
\][/tex]
2. Rewrite the inequality with the expanded expressions:
[tex]\[
-6x - 12 \geq 4x + 28
\][/tex]
3. Isolate the variable [tex]\(x\)[/tex] on one side of the inequality:
- Start by adding [tex]\(6x\)[/tex] to both sides:
[tex]\[
-12 \geq 10x + 28
\][/tex]
- Next, subtract [tex]\(28\)[/tex] from both sides:
[tex]\[
-40 \geq 10x
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
- Divide both sides by [tex]\(10\)[/tex]:
[tex]\[
-4 \geq x
\][/tex]
- This can also be written as:
[tex]\[
x \leq -4
\][/tex]
So, the inequality that represents all the solutions is [tex]\(x \leq -4\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{B. \, x \leq -4} \][/tex]
