https://file.aiquickdraw.com/mathbot/file/17273712993389430lqli.jpg

To define the shaded region in the graph, we need to derive the equations of the three boundary lines, then convert those into inequalities.

### Step 1: Identify the lines

1. **Line 1** (Upper slanting line):
   - Passes through points \((0, 2)\) and \((6, 6)\).
   - Slope \(m = \frac{6 - 2}{6 - 0} = \frac{4}{6} = \frac{2}{3}\).
   - Using the point-slope form: \(y - 2 = \frac{2}{3}(x - 0)\).
   - The equation of the line is: \(y = \frac{2}{3}x + 2\).

2. **Line 2** (Lower slanting line):
   - Passes through points \((0, 0)\) and \((6, -6)\).
   - Slope \(m = \frac{-6 - 0}{6 - 0} = -1\).
   - Using the point-slope form: \(y - 0 = -1(x - 0)\).
   - The equation of the line is: \(y = -x\).

3. **Line 3** (Vertical line):
   - A vertical line at \(x = 0\).

### Step 2: Write the inequalities

The shaded region is enclosed by these three lines. The inequalities must capture this region:

1. For **Line 1**: The region lies below the line \(y = \frac{2}{3}x + 2\).
   - Inequality: \(y \leq \frac{2}{3}x + 2\).

2. For **Line 2**: The region lies above the line \(y = -x\).
   - Inequality: \(y \geq -x\).

3. For **Line 3**: The region lies to the right of the vertical line \(x = 0\).
   - Inequality: \(x \geq 0\).

### Final Answer:
The three inequalities defining the shaded region are:
\[
y \leq \frac{2}{3}x + 2
\]
\[
y \geq -x
\]
\[
x \geq 0
\]

Would you like further details or have any other questions?

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Here are 5 related questions to extend this concept:
1. How would the inequalities change if the shaded region were above both lines?
2. Can you determine the area of the shaded region?
3. What would happen if one of the lines was shifted? How would the inequalities change?
4. How would you solve for the points of intersection between the boundary lines?
5. What is the slope of a line perpendicular to each of these boundary lines?

**Tip:** When converting a line equation to an inequality, always check if the shading is above or below the line.

Write down the three inequalities that define the shaded region.

Learn how to find the inequalities defining a shaded region in a coordinate plane. This problem covers key concepts such as calculating the slope of lines, finding the equation of lines, and converting them into inequalities. Suitable for students in Grades 9-11.

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