To determine which value of [tex]\( x \)[/tex] is in the solution set for the inequality [tex]\( 3(x - 4) \geq 5x + 2 \)[/tex], we can follow a step-by-step approach to solve the inequality.
1. Distribute the 3 on the left side:
[tex]\[
3(x - 4) = 3x - 12
\][/tex]
So, the inequality becomes:
[tex]\[
3x - 12 \geq 5x + 2
\][/tex]
2. Move all terms involving [tex]\( x \)[/tex] to one side and constants to the other:
[tex]\[
3x - 12 \geq 5x + 2
\][/tex]
Subtract [tex]\( 5x \)[/tex] from both sides:
[tex]\[
3x - 5x - 12 \geq 2
\][/tex]
Simplifying gives:
[tex]\[
-2x - 12 \geq 2
\][/tex]
3. Isolate the term with [tex]\( x \)[/tex]:
Add 12 to both sides:
[tex]\[
-2x - 12 + 12 \geq 2 + 12
\][/tex]
Simplifying gives:
[tex]\[
-2x \geq 14
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Divide both sides by [tex]\(-2\)[/tex] (remember, dividing by a negative number reverses the inequality sign):
[tex]\[
x \leq \frac{14}{-2}
\][/tex]
Simplifying gives:
[tex]\[
x \leq -7
\][/tex]
Now, we need to check which of the given options satisfy [tex]\( x \leq -7 \)[/tex].
- For [tex]\( x = -10 \)[/tex]:
[tex]\[
-10 \leq -7 \text{ (True)}
\][/tex]
- For [tex]\( x = -5 \)[/tex]:
[tex]\[
-5 \leq -7 \text{ (False)}
\][/tex]
- For [tex]\( x = 5 \)[/tex]:
[tex]\[
5 \leq -7 \text{ (False)}
\][/tex]
- For [tex]\( x = 10 \)[/tex]:
[tex]\[
10 \leq -7 \text{ (False)}
\][/tex]
Only [tex]\( x = -10 \)[/tex] satisfies the inequality [tex]\( x \leq -7 \)[/tex].
Therefore, the value of [tex]\( x \)[/tex] that is in the solution set of [tex]\( 3(x-4) \geq 5x+2 \)[/tex] is:
[tex]\[
\boxed{-10}
\][/tex]
