Let's solve the given compound inequality step-by-step.
### Step 1: Solve the first inequality [tex]\(6b < 42\)[/tex].
To isolate [tex]\(b\)[/tex], divide both sides of the inequality by 6:
[tex]\[ \frac{6b}{6} < \frac{42}{6} \][/tex]
[tex]\[ b < 7 \][/tex]
### Step 2: Solve the second inequality [tex]\(4b + 12 > 8\)[/tex].
First, subtract 12 from both sides to isolate the term with [tex]\(b\)[/tex]:
[tex]\[ 4b + 12 - 12 > 8 - 12 \][/tex]
[tex]\[ 4b > -4 \][/tex]
Next, divide both sides by 4:
[tex]\[ \frac{4b}{4} > \frac{-4}{4} \][/tex]
[tex]\[ b > -1 \][/tex]
### Step 3: Combine the inequalities.
The compound inequality is:
[tex]\[ b < 7 \quad \text{or} \quad b > -1 \][/tex]
Therefore, the solution to the compound inequality [tex]\(6b < 42\)[/tex] or [tex]\(4b + 12 > 8\)[/tex] is:
[tex]\[ b < 7 \quad \text{or} \quad b > -1 \][/tex]
### Conclusion:
Among the provided options, the correct one is:
[tex]\[ b < 7 \quad \text{or} \quad b > -1 \][/tex]
So, the correct choice is:
[tex]\[ b < 7 \quad \text{or} \quad b > -1 \][/tex]
