Let's solve the inequality [tex]\( -b \leq -2(3 + b) \)[/tex] step by step.
1. Distribute the [tex]\(-2\)[/tex] inside the parentheses on the right side:
[tex]\[
-b \leq -2 \cdot 3 + (-2) \cdot b \implies -b \leq -6 - 2b
\][/tex]
2. Isolate [tex]\(b\)[/tex] on one side. To do this, add [tex]\(2b\)[/tex] to both sides of the inequality:
[tex]\[
-b + 2b \leq -6 - 2b + 2b \implies b \leq -6
\][/tex]
So far we have determined:
[tex]\[
b \leq -6
\][/tex]
Since there are no further terms to isolate or simplify, this is our solution expressed in interval notation.
Therefore, [tex]\( -\infty < b \leq -6 \)[/tex]. Looking for specific values that satisfy this inequality from the given choices:
Given choices:
1. [tex]\(0\)[/tex]
2. [tex]\(-6\)[/tex]
3. [tex]\(6\)[/tex]
4. [tex]\(-12\)[/tex]
5. [tex]\(-1\)[/tex]
The values that satisfy [tex]\(b \leq -6\)[/tex] from the given options are:
- [tex]\( -6 \)[/tex] (since [tex]\(-6 = -6\)[/tex])
- [tex]\( -12 \)[/tex] (since [tex]\(-12 < -6\)[/tex])
Thus, the values that satisfy the inequality are:
[tex]\[
-6 \quad \text{and} \quad -12
\][/tex]
