To graph linear inequalities, follow these steps:

### Steps for Graphing Linear Inequalities:

1. **Rewrite the inequality (if needed):**
   Ensure the inequality is in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The inequality will look like:
   \[
   y < mx + b, \quad y \leq mx + b, \quad y > mx + b, \quad \text{or} \quad y \geq mx + b
   \]
   If it's not in this form, solve for \(y\).

2. **Graph the boundary line:**
   - First, graph the line as if it were an equality (e.g., for \(y \leq mx + b\), graph \(y = mx + b\)).
   - If the inequality is strict (\(<\) or \(>\)), use a **dashed** line to indicate that points on the line are not included in the solution.
   - If the inequality is inclusive (\(\leq\) or \(\geq\)), use a **solid** line to include the points on the line.

3. **Test a point:**
   Choose a test point not on the boundary line (commonly \( (0,0) \), unless it’s on the line). Plug this point into the inequality:
   - If it satisfies the inequality, shade the region that includes the test point.
   - If it does not satisfy the inequality, shade the opposite side of the line.

4. **Shade the correct region:**
   - The region of the graph that satisfies the inequality is where you shade.
   - For example, if the inequality is \( y > mx + b \), you would shade **above** the line. If the inequality is \( y < mx + b \), shade **below** the line.

5. **Label the graph:**
   Ensure the axes, the boundary line, and shaded region are properly labeled to clearly represent the inequality.

### Example:
For the inequality \( y \leq 2x - 3 \):
- Graph the line \( y = 2x - 3 \) as a **solid** line.
- Use a test point like \( (0, 0) \):
   \[
   0 \leq 2(0) - 3 \quad \Rightarrow \quad 0 \leq -3 \quad \text{(false)}
   \]
  Since it’s false, shade the side of the line that **does not** include \( (0,0) \).

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### 5 Related Questions:
1. How do you graph systems of linear inequalities?
2. What are some real-world applications of linear inequalities in graphing?
3. How can you graph inequalities in 3D (with two variables)?
4. How do you find the vertices of a region defined by inequalities?
5. Can linear inequalities represent non-straight boundaries?

**Tip:** Always choose simple test points (like \( (0,0) \)) when it's not on the boundary, to make checking easy.

Learn how to graph linear inequalities using simple steps. This tutorial covers key algebra concepts, including slope-intercept form and testing points. Suitable for students in Grades 8-10, this guide provides a clear method to visualize and solve inequalities.

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