Question

Solve the compound inequality.
6b - 1 less than or equal to 213 and 2b + 1 greater than or equal to 11

Answer 1

To solve the compound inequality \(6b - 1 \leq 213\) and \(2b + 1 \geq 11\), we can solve each part separately.

1. **Solve \(6b - 1 \leq 213\)**:

\[
6b - 1 \leq 213
\]

Add 1 to both sides:

\[
6b \leq 214
\]

Now, divide both sides by 6:

\[
b \leq \frac{214}{6}
\]

Simplify:

\[
b \leq \frac{107}{3} \quad \text{(approximately 35.67)}
\]

2. **Solve \(2b + 1 \geq 11\)**:

\[
2b + 1 \geq 11
\]

Subtract 1 from both sides:

\[
2b \geq 10
\]

Now, divide both sides by 2:

\[
b \geq 5
\]

Now we have the two inequalities:

\[
5 \leq b \leq \frac{107}{3}
\]

Thus, the solution to the compound inequality is:

\[
b \in [5, \frac{107}{3}]
\]

or in interval notation,

\[
[5, \frac{107}{3}]
\]