The inequality given is:

\[
\frac{3x + 5}{x - 3} \leq \frac{1}{2}
\]

### Step 1: Eliminate the denominator by cross-multiplying

We need to multiply both sides of the inequality by \( 2(x - 3) \). However, since \( x - 3 \) can be either positive or negative depending on the value of \( x \), we must treat this carefully.

- If \( x - 3 > 0 \) (i.e., \( x > 3 \)), we can simply multiply without changing the inequality sign.
- If \( x - 3 < 0 \) (i.e., \( x < 3 \)), multiplying by a negative number will reverse the inequality sign.

Let’s proceed with two cases.

### Case 1: \( x > 3 \)

We can multiply both sides by \( 2(x - 3) \), and the inequality becomes:

\[
2(3x + 5) \leq (x - 3)
\]

Expanding both sides:

\[
6x + 10 \leq x - 3
\]

Subtract \( x \) from both sides:

\[
5x + 10 \leq -3
\]

Subtract 10 from both sides:

\[
5x \leq -13
\]

Divide by 5:

\[
x \leq -\frac{13}{5}
\]

But remember, this case assumes \( x > 3 \), and \( x \leq -\frac{13}{5} \) does not satisfy \( x > 3 \), so there is no solution in this case.

### Case 2: \( x < 3 \)

Now we multiply both sides by \( 2(x - 3) \), but this time we reverse the inequality sign because \( x - 3 \) is negative for \( x < 3 \):

\[
2(3x + 5) \geq (x - 3)
\]

Expanding both sides:

\[
6x + 10 \geq x - 3
\]

Subtract \( x \) from both sides:

\[
5x + 10 \geq -3
\]

Subtract 10 from both sides:

\[
5x \geq -13
\]

Divide by 5:

\[
x \geq -\frac{13}{5}
\]

Since this case assumes \( x < 3 \), the solution is:

\[
-\frac{13}{5} \leq x < 3
\]

### Final Answer:

The solution to the inequality is:

\[
-\frac{13}{5} \leq x < 3
\]

### Do you want further details or have any questions?

Here are some related questions you could explore:
1. How do you solve a quadratic inequality?
2. What happens when cross-multiplying with inequalities?
3. How can you graph the solution of this inequality?
4. How do you test whether a boundary point should be included in an inequality solution?
5. What is the difference between a strict inequality and a non-strict inequality?

**Tip:** Always be careful when multiplying or dividing inequalities by negative numbers since it reverses the inequality sign!

Explore the step-by-step solution to the rational inequality 3x + 5 / (x - 3) ≤ 1/2. This problem demonstrates the process of cross-multiplication, analyzing cases where the denominator changes sign, and solving linear inequalities. Suitable for high school students in Grades 10-12.

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