Solve for s and graph the solution.
6>
5s+3

3
>–
4
Plot the endpoints. Select an endpoint to change it from closed to open. Select the middle of a segment, ray, or line to delete it.

To solve the compound inequality and graph the solution, let's start by interpreting and solving each part of the compound inequality.

### Given Compound Inequality:
\[
6 > \frac{5s + 3}{4} > 3
\]

### Step 1: Break Down the Compound Inequality

This compound inequality can be split into two separate inequalities:

1. \( 6 > \frac{5s + 3}{4} \)
2. \( \frac{5s + 3}{4} > 3 \)

### Step 2: Solve Each Inequality

#### First Inequality:
\[
6 > \frac{5s + 3}{4}
\]

Multiply both sides by 4 to eliminate the fraction:

\[
24 > 5s + 3
\]

Subtract 3 from both sides:

\[
21 > 5s
\]

Divide by 5:

\[
\frac{21}{5} > s
\]

Or equivalently:

\[
s < \frac{21}{5}
\]

This simplifies to:

\[
s < 4.2
\]

#### Second Inequality:
\[
\frac{5s + 3}{4} > 3
\]

Multiply both sides by 4 to eliminate the fraction:

\[
5s + 3 > 12
\]

Subtract 3 from both sides:

\[
5s > 9
\]

Divide by 5:

\[
s > \frac{9}{5}
\]

This simplifies to:

\[
s > 1.8
\]

### Step 3: Combine the Results
The solution to the compound inequality is:

\[
1.8 < s < 4.2
\]

### Step 4: Graph the Solution

On a number line:
- Plot an **open dot** at \( s = 1.8 \), indicating \( s > 1.8 \).
- Plot an **open dot** at \( s = 4.2 \), indicating \( s < 4.2 \).
- Draw a line connecting the open dots between 1.8 and 4.2 to represent the interval \( (1.8, 4.2) \).

Here is how it would look on a number line:

```
---o====================o---
   1.8                  4.2
```

**Would you like further details or a graph for this?**

---

### Related Questions:
1. How do you solve compound inequalities involving "or" conditions?
2. How would the solution change if one of the inequalities had a "greater than or equal to" symbol?
3. How do you solve inequalities when the variable is in the denominator?
4. What steps should be taken if the compound inequality involves absolute values?
5. How can you check if your solution to a compound inequality is correct?

**Tip:** Always double-check whether the inequality signs are strict (>) or inclusive (≥) to ensure that the correct type of dot (open or closed) is used on the number line.

Learn how to solve the compound inequality 6 > (5s + 3)/4 > 3 step-by-step and graph the solution on a number line. This problem covers compound inequalities, number line graphing, and includes detailed solution steps. Suitable for Grades 9-12 students.

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