To graph the inequality \( 8x + 2y > 10 \), we need to follow these steps:

### Step 1: Rewrite the inequality
First, solve for \( y \) to express the inequality in a more graphable form:

\[
8x + 2y > 10
\]

Subtract \( 8x \) from both sides:

\[
2y > -8x + 10
\]

Now, divide everything by 2 to solve for \( y \):

\[
y > -4x + 5
\]

### Step 2: Graph the boundary line
The boundary line corresponds to the equation \( y = -4x + 5 \). This is a straight line with:
- **Slope**: \( -4 \)
- **Y-intercept**: \( 5 \)

To plot this line:
1. Start at the y-intercept, which is \( (0, 5) \).
2. Use the slope \( -4 \), which means for every 1 unit increase in \( x \), \( y \) decreases by 4 units. So from \( (0, 5) \), move to \( (1, 1) \).
3. Connect these points with a straight line.

This line will be **dashed** because the inequality is strict (\( > \)) and not \( \geq \).

### Step 3: Shade the correct region
Since the inequality is \( y > -4x + 5 \), we shade **above** the line, as we are looking for all points where \( y \) is greater than \( -4x + 5 \).

### Visualization:
You would graph a dashed line through points like \( (0, 5) \) and \( (1, 1) \), then shade the area above this line.

Would you like me to generate a graph for this, or do you have any further questions?

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Here are 5 related questions:
1. How would the graph change if the inequality was \( y \geq -4x + 5 \)?
2. What is the significance of the slope in the inequality?
3. How can we find the x-intercept of the boundary line?
4. How does changing the coefficient of \( x \) affect the graph?
5. What happens if we reverse the inequality to \( y < -4x + 5 \)?

**Tip:** Always use a dashed line for strict inequalities (like \( > \) or \( < \)) and a solid line for inclusive inequalities (like \( \geq \) or \( \leq \)).

Learn how to graph the linear inequality 8x + 2y > 10 by transforming it into slope-intercept form, graphing the boundary line, and shading the correct region. This guide covers core algebraic concepts such as slope, y-intercept, and graphing inequalities, suitable for students in Grades 8-10.

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