To solve the compound inequality [tex]\(6b < 42\)[/tex] or [tex]\(4b + 12 > 8\)[/tex], let's tackle each inequality step-by-step.
1. Solve the first inequality [tex]\(6b < 42\)[/tex]:
- Divide both sides by 6 to isolate [tex]\(b\)[/tex]:
[tex]\[
b < \frac{42}{6}
\][/tex]
- This simplifies to:
[tex]\[
b < 7
\][/tex]
2. Solve the second inequality [tex]\(4b + 12 > 8\)[/tex]:
- Start by isolating the term with [tex]\(b\)[/tex] on one side. Subtract 12 from both sides:
[tex]\[
4b > 8 - 12
\][/tex]
- This simplifies to:
[tex]\[
4b > -4
\][/tex]
- Now divide both sides by 4 to solve for [tex]\(b\)[/tex]:
[tex]\[
b > \frac{-4}{4}
\][/tex]
- This simplifies to:
[tex]\[
b > -1
\][/tex]
So the solutions to the individual inequalities are:
- [tex]\(b < 7\)[/tex]
- [tex]\(b > -1\)[/tex]
When we combine these solutions using "or", we get:
[tex]\[ b < 7 \text{ or } b > -1 \][/tex]
Therefore, the correct answer is:
[tex]\[ b < 7 \text{ or } b > -1 \][/tex]
