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!

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.