To solve the inequality [tex]\(-3(x-6) > 2x - 2\)[/tex], we will go through it step by step.
1. Distribute the [tex]\(-3\)[/tex] on the left side:
[tex]\[
-3(x - 6) = -3 \cdot x + -3 \cdot (-6) = -3x + 18
\][/tex]
So the inequality becomes:
[tex]\[
-3x + 18 > 2x - 2
\][/tex]
2. Move all terms involving [tex]\(x\)[/tex] to one side and constants to the other:
Add [tex]\(3x\)[/tex] to both sides:
[tex]\[
18 > 5x - 2 + 3x
\][/tex]
Simplify:
[tex]\[
18 > 5x - 2
\][/tex]
3. Add 2 to both sides to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[
18 + 2 > 5x
\][/tex]
Simplify:
[tex]\[
20 > 5x
\][/tex]
4. Divide both sides by 5 to solve for [tex]\(x\)[/tex]:
[tex]\[
\frac{20}{5} > x
\][/tex]
Simplify:
[tex]\[
4 > x
\][/tex]
Or equivalently,
[tex]\[
x < 4
\][/tex]
Thus, the solution to [tex]\(-3(x-6) > 2 x - 2\)[/tex] is:
[tex]\[
x < 4
\][/tex]
Therefore, the correct answer is:
[tex]\[
(2) \, x < 4
\][/tex]
