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The given problem consists of solving the following compound inequality:

\[
\frac{11 + 3x}{5} \geq 3 - x > -\frac{3}{2}, \quad x \in \mathbb{R}
\]

### Step-by-Step Solution

#### Step 1: Break the compound inequality into two parts.
We will solve it by first splitting the inequality into two parts:

1. \(\frac{11 + 3x}{5} \geq 3 - x\)
2. \(3 - x > -\frac{3}{2}\)

#### Step 2: Solve the first inequality \(\frac{11 + 3x}{5} \geq 3 - x\)

\[
\frac{11 + 3x}{5} \geq 3 - x
\]

Multiply both sides of the inequality by 5 to eliminate the denominator:

\[
11 + 3x \geq 5(3 - x)
\]

Expand the right-hand side:

\[
11 + 3x \geq 15 - 5x
\]

Now, combine like terms:

\[
3x + 5x \geq 15 - 11
\]

\[
8x \geq 4
\]

Now, divide both sides by 8:

\[
x \geq \frac{4}{8} = \frac{1}{2}
\]

#### Step 3: Solve the second inequality \(3 - x > -\frac{3}{2}\)

\[
3 - x > -\frac{3}{2}
\]

Subtract 3 from both sides:

\[
-x > -\frac{3}{2} - 3
\]

Convert 3 to a fraction with a denominator of 2:

\[
-x > -\frac{3}{2} - \frac{6}{2} = -\frac{9}{2}
\]

Now, multiply both sides by -1, remembering to flip the inequality sign:

\[
x < \frac{9}{2}
\]

#### Step 4: Combine the two results

From Step 2, we have \(x \geq \frac{1}{2}\), and from Step 3, we have \(x < \frac{9}{2}\). Combining these gives the solution:

\[
\frac{1}{2} \leq x < \frac{9}{2}
\]

### Step 5: Write the solution set

The solution set is:

\[
x \in \left[\frac{1}{2}, \frac{9}{2}\right)
\]

### Step 6: Represent the solution on the number line

This interval represents all values of \(x\) between \(\frac{1}{2}\) and \(\frac{9}{2}\), including \(\frac{1}{2}\) but not \(\frac{9}{2}\). You can draw a number line with a filled circle at \(\frac{1}{2}\) and an open circle at \(\frac{9}{2}\), shading the region in between.

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Would you like more details or have any questions about the process? Here are 5 related questions to explore further:

1. How do you solve compound inequalities involving fractions?
2. What is the importance of reversing the inequality when multiplying by a negative number?
3. How would the solution change if the inequality signs were reversed?
4. How can inequalities be solved graphically on a coordinate plane?
5. What are the common mistakes to avoid when solving multi-step inequalities?

**Tip:** Always remember to flip the inequality sign when multiplying or dividing by a negative number during your calculations.

Solve the inequality: (11 + 3x) / 5 ≥ 3 - x > -3/2, x ∈ R. (a) Write the solution set. (b) Represent the solution on the number line.

Learn how to solve the compound inequality (11 + 3x) / 5 ≥ 3 - x > -3/2. The solution set is provided along with step-by-step instructions and a number line representation. This problem covers key algebraic concepts such as isolating variables in inequalities and applying properties of inequalities. Suitable for Grades 10-12 students.

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