https://file.aiquickdraw.com/mathbot/file/1727366025860u4x1zp17.jpg

The question asks to shade the region that satisfies three inequalities, which are:

1. \( y \leq -3x + 6 \)
2. \( y + x \geq 0 \) (which simplifies to \( y \geq -x \))
3. \( y < 6 \)

### Step-by-Step Process:

1. **Inequality 1: \( y \leq -3x + 6 \)**  
   This is the equation of a straight line \( y = -3x + 6 \) with a slope of -3 and a y-intercept of 6. The shaded region is below this line because \( y \) is less than or equal to the line.

2. **Inequality 2: \( y \geq -x \)**  
   This is another straight line equation \( y = -x \), which passes through the origin with a slope of -1. The region above this line should be shaded as \( y \) is greater than or equal to \( -x \).

3. **Inequality 3: \( y < 6 \)**  
   This inequality indicates that the y-values must be strictly less than 6. This forms a horizontal boundary at \( y = 6 \), so the region below this line (but not including it) should be shaded.

### Shading the Region:
To find the solution, we will need to identify the region that satisfies all three conditions simultaneously. This region is the area where all shaded regions from the inequalities overlap. Specifically:

- The region is below the line \( y = -3x + 6 \).
- It is above or on the line \( y = -x \).
- It is strictly below the horizontal line \( y = 6 \).

### Graphically:
- The line \( y = -3x + 6 \) intersects the y-axis at \( (0, 6) \) and the x-axis at \( (2, 0) \).
- The line \( y = -x \) passes through the origin \( (0, 0) \) and intersects the graph at \( (-1, 1) \) and other points.

Thus, the region where all conditions meet is a triangular region, starting from \( (0, 0) \), with boundaries formed by these lines and restricted below \( y = 6 \).

Would you like a detailed drawing of this region or further clarification? Let me know if you have any specific questions!

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Here are 5 related questions for practice:
1. How would the solution change if the inequality \( y + x \geq 0 \) were replaced by \( y + x \leq 0 \)?
2. What is the significance of the strict inequality \( y < 6 \) in this problem?
3. How can you determine the points of intersection between the lines \( y = -3x + 6 \) and \( y = -x \)?
4. How would you solve the inequalities if given only two instead of three?
5. Can you explain how the graph might look for different values of \( x \) for each line?

**Tip:** When solving inequalities graphically, always sketch the lines first, then decide which side of the line corresponds to the solution region.

In the graph below, shade the region which satisfies the three inequalities y ≤ -3x + 6, y + x ≥ 0, and y < 6.

This tutorial explains how to solve and shade the region that satisfies the inequalities y ≤ -3x + 6, y + x ≥ 0, and y < 6. It includes detailed steps to graph the inequalities, find the intersection points, and identify the solution region. Perfect for students in Grades 10-12.

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