Solve for p and graph the solution.
7p–2

4
≥3 or 10p–9≤1
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, we need to address each inequality separately:

### Given Inequalities:
1. \( \frac{7p - 2}{4} \geq 3 \)
2. \( 10p - 9 \leq 1 \)

### Step 1: Solve the first inequality

\[
\frac{7p - 2}{4} \geq 3
\]

Multiply both sides by 4 to eliminate the fraction:

\[
7p - 2 \geq 12
\]

Add 2 to both sides:

\[
7p \geq 14
\]

Divide by 7:

\[
p \geq 2
\]

### Step 2: Solve the second inequality

\[
10p - 9 \leq 1
\]

Add 9 to both sides:

\[
10p \leq 10
\]

Divide by 10:

\[
p \leq 1
\]

### Step 3: Combine and interpret the solution
- The first inequality \( p \geq 2 \) indicates \( p \) is greater than or equal to 2.
- The second inequality \( p \leq 1 \) indicates \( p \) is less than or equal to 1.

### Step 4: Graph the solution

These inequalities describe two different regions:
- \( p \geq 2 \) represents the interval \([2, \infty)\) on the number line.
- \( p \leq 1 \) represents the interval \((-\infty, 1]\) on the number line.

Since the solution is an "or" inequality, it includes all values where \( p \) satisfies either inequality:

**Final Answer:**

\[
p \in (-\infty, 1] \cup [2, \infty)
\]

### Step 5: Graphing
On the number line:
- Plot a **closed dot** at \( p = 1 \), indicating \( p \leq 1 \).
- Plot a **closed dot** at \( p = 2 \), indicating \( p \geq 2 \).
- Draw a ray extending leftward from \( p = 1 \) to represent \( p \leq 1 \).
- Draw a ray extending rightward from \( p = 2 \) to represent \( p \geq 2 \).

Here is how it would look on a number line:

```
<---●====================|--------------------●===>
     -∞              1                     2     +∞
```

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

---

### Related Questions:
1. How do you solve compound inequalities involving "and" conditions?
2. How would the solution change if the inequalities were connected by "and" instead of "or"?
3. What does an open dot on a number line represent compared to a closed dot?
4. How can we graph inequalities that involve absolute values?
5. How do we approach inequalities with variables on both sides?

**Tip:** When solving inequalities, always pay close attention to whether the inequality symbol includes an equality (≥, ≤) or not (> , <), as this affects how you graph the solution.

Learn how to solve compound inequalities involving graphical representation. This problem covers solving inequalities, plotting endpoints on a number line, and interpreting solution intervals. Suitable for high school students studying algebra.

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