https://file.aiquickdraw.com/mathbot/file/1728254317576fqtux56w.jpg

Let's graph the inequality given in the image: 

### Inequality: 
\[ 5x - y > 10 \]

To graph this inequality, we first need to transform it into a form that helps us determine the boundary line and shading. Here's how we do it:

1. **Rewrite in Slope-Intercept Form**:
   Start by isolating \(y\):
   \[
   5x - y > 10
   \]
   Subtract \(5x\) from both sides:
   \[
   -y > -5x + 10
   \]
   Multiply by \(-1\) (and remember to flip the inequality sign):
   \[
   y < 5x - 10
   \]

2. **Graph the Boundary Line**:
   - The boundary line for this inequality is \(y = 5x - 10\).
   - This line will be **dashed**, not solid, because the inequality is strict (i.e., \(<\) and not \(\leq\)).
   - Plot the y-intercept at \((0, -10)\).
   - Use the slope, which is \(5\), meaning the line goes up 5 units for every 1 unit to the right. You can plot a second point using this slope, e.g., at \((1, -5)\).

3. **Shading the Region**:
   - Since the inequality is \(y < 5x - 10\), shade **below** the line. This indicates that the values for \(y\) are less than the boundary line.

### Steps for Graphing:
- **Plot the line** \(y = 5x - 10\) with a **dashed line** to indicate that points on the line are not included in the solution.
- **Shade below** the dashed line to represent all the points \((x, y)\) that satisfy the inequality.

Would you like me to go into further detail about graphing inequalities, or do you have any specific questions about this problem?

Here are 5 related questions to expand your understanding:

1. How would the graph change if the inequality were \(5x - y \geq 10\)?
2. Can you determine whether the point \((3, 2)\) satisfies the given inequality?
3. What is the significance of a solid line versus a dashed line in graphing inequalities?
4. How would you write the inequality if the region above the line were shaded instead?
5. How can you test a point to confirm which side of the line to shade?

**Tip**: When graphing inequalities, always choose a test point (like \((0,0)\)) to verify which region should be shaded—especially if you're unsure where the boundary lies!

Learn how to graph the inequality 5x - y > 10. This problem explores the conversion of a linear inequality to slope-intercept form and the process of graphing with a dashed line and shading. Suitable for students in Grades 8-10.

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