How can I know which region should I shade ?

To determine which region to shade for a given inequality, you can follow these steps:

### 1. **Understand the Inequality Sign**
   - **\(<\) or \(>\):** These signs indicate that the inequality is *strict*, meaning the boundary line itself is **not included** in the solution. The line will be **dashed**.
   - **\(\leq\) or \(\geq\):** These signs indicate that the inequality is *inclusive*, meaning the boundary line **is included** in the solution. The line will be **solid**.

### 2. **Identify the Boundary Line**
   - Convert the inequality into an equation (e.g., \(y \leq 2x + 1\) becomes \(y = 2x + 1\)).
   - Plot this line on the coordinate plane.

### 3. **Choose a Test Point**
   - Pick a test point that is not on the boundary line (usually, the origin \((0, 0)\) is a convenient choice unless the line passes through it).
   - Substitute this point into the original inequality.
   
   **Example:**
   - Inequality: \(y \leq 2x + 1\)
   - Test point: \((0, 0)\)
   - Substitute: \(0 \leq 2(0) + 1 \implies 0 \leq 1\) (True)

### 4. **Determine the Shaded Region**
   - If the test point **satisfies** the inequality (i.e., the inequality holds true), then the region containing the test point is the **solution region**, and you should shade that region.
   - If the test point **does not satisfy** the inequality (i.e., the inequality does not hold true), then the solution region is on the opposite side of the line, and you should shade that region.

### 5. **Check the Shading Against Multiple Inequalities**
   - When dealing with a system of inequalities, repeat the steps for each inequality.
   - The solution to the system will be the **intersection** of all the shaded regions. Only shade the area where all inequalities overlap.

### **Example Walkthrough:**

Consider the inequality **\(y \geq x - 2\)**.

1. **Convert to the equation:** \(y = x - 2\).
2. **Plot the boundary line:** This is a line with a slope of 1 and a y-intercept of -2. Since the inequality is \(\geq\), the line is solid.
3. **Choose a test point:** Let's use \((0, 0)\).
4. **Substitute the test point:** 
   - \(0 \geq 0 - 2 \implies 0 \geq -2\) (True)
5. **Shade the region:** Since the test point (0, 0) satisfies the inequality, shade the region **above the line** \(y = x - 2\).

### Tips for Shading:
- If \(y\) is isolated and you have \(y <\) or \(y >\), shade **below** the line for \(y <\) and **above** the line for \(y >\).
- Always use a test point to verify which side of the line to shade.
- For systems of inequalities, shade the overlapping region that satisfies all inequalities.

Would you like to practice with another example or have more questions?

Here are some additional questions for further understanding:
1. How do you shade the region for horizontal and vertical lines?
2. What happens if you choose a test point that lies exactly on the boundary line?
3. How can graphing software or tools help visualize inequalities?
4. How do you handle shading when dealing with inequalities that involve \(x\) only, like \(x > 2\)?
5. How would inequalities look on a graph in three dimensions?

**Tip:** Always draw the boundary line first and clearly before determining which side to shade to avoid confusion.

Learn how to determine which region to shade for inequalities with a detailed step-by-step guide. Understand the signs of inequalities, plot boundary lines, choose test points, and shade the correct regions based on the test results. Suitable for high school students and anyone learning about inequalities in mathematics.

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