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Let's first solve the inequality step by step.

The inequality is given as:
\[
\frac{5}{3}(b - 12) \geq \frac{3}{2}(7b - 11)
\]

### Step 1: Eliminate the fractions
To eliminate the fractions, we can multiply both sides of the inequality by 6, which is the least common denominator of 3 and 2.

Multiplying both sides by 6:
\[
6 \times \frac{5}{3}(b - 12) \geq 6 \times \frac{3}{2}(7b - 11)
\]
Simplifying:
\[
2 \times 5(b - 12) \geq 3 \times 3(7b - 11)
\]
\[
10(b - 12) \geq 9(7b - 11)
\]

### Step 2: Expand both sides
Now, expand both sides:
\[
10b - 120 \geq 63b - 99
\]

### Step 3: Move the variable terms to one side and the constant terms to the other side
Let's subtract \(10b\) from both sides:
\[
-120 \geq 53b - 99
\]

Now, add 99 to both sides:
\[
-120 + 99 \geq 53b
\]
\[
-21 \geq 53b
\]

### Step 4: Solve for \(b\)
To isolate \(b\), divide both sides by 53:
\[
b \leq \frac{-21}{53}
\]

### Step 5: Solution in different notations

- **Set-builder notation**: \(\{b \mid b \leq \frac{-21}{53} \}\)
- **Interval notation**: \(\left(-\infty, \frac{-21}{53}\right]\)

### Step 6: Graph the solution

We are looking for the graph that shows all values of \(b\) less than or equal to \(\frac{-21}{53}\), which is slightly less than 0. This corresponds to a closed circle at \(\frac{-21}{53}\) and a shaded line extending to the left. Based on this, **Graph B** is the correct answer.

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Would you like more details on any part of this solution?

Here are 5 related questions you could explore:
1. How do you handle inequalities with fractions?
2. What is the significance of the closed vs. open circle in graphing inequalities?
3. How do you convert between set-builder notation and interval notation?
4. How does multiplying or dividing both sides of an inequality by a negative number affect the inequality?
5. Can inequalities with absolute values be solved similarly?

**Tip:** Always remember to flip the inequality sign when multiplying or dividing both sides by a negative number!

Solve the linear inequality and choose the correct graph: (5/3)(b - 12) ≥ (3/2)(7b - 11).

Learn how to solve the linear inequality (5/3)(b - 12) ≥ (3/2)(7b - 11) with a step-by-step solution. This problem involves key algebraic concepts such as eliminating fractions and solving linear inequalities. Suitable for students in Grades 9-10.

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