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The inequality you are working with is:

\[
y \geq -\frac{1}{2}x + 7
\]

This is a linear inequality, and to graph it, we need to follow these steps:

### Step 1: Graph the boundary line
The boundary line for the inequality is:
\[
y = -\frac{1}{2}x + 7
\]

This is a straight line with:
- A slope of \(-\frac{1}{2}\), meaning it falls 1 unit vertically for every 2 units it moves horizontally.
- A y-intercept of 7, meaning the line crosses the y-axis at the point \((0, 7)\).

Using this information:
- Start by plotting the point \((0, 7)\).
- From there, apply the slope: move 2 units to the right and 1 unit down to get another point on the line (e.g., \((2, 6)\)).
- Draw a straight line passing through these points.

### Step 2: Determine the type of line
Since the inequality is \(\geq\) (greater than or equal to), the boundary line is **solid**, indicating that points on the line are included in the solution.

### Step 3: Shade the appropriate region
For the inequality \(y \geq -\frac{1}{2}x + 7\):
- Shade the region **above** the line, because we are dealing with \(y\)-values that are greater than or equal to the line's equation.

### Summary:
- Plot a solid line passing through \((0, 7)\) with a slope of \(-\frac{1}{2}\).
- Shade the area above the line.

Do you have any questions about this, or would you like me to break down any of the steps further?

Here are 5 related questions for further practice:
1. How would the graph change if the inequality were \(y < -\frac{1}{2}x + 7\)?
2. What is the equation of the boundary line if the slope is changed to \(\frac{1}{2}\)?
3. How do you graph \(y \leq 3x - 2\)?
4. How do you determine whether to shade above or below the boundary line for linear inequalities?
5. How would you graph a system of two linear inequalities?

**Tip**: Always check the inequality symbol to decide whether the boundary line should be solid or dashed.

Learn how to graph the linear inequality y ≥ -1/2x + 7 by plotting the boundary line and shading the appropriate region. Understand the steps involved in determining the slope, y-intercept, and graphing solid boundary lines. Suitable for high school algebra students in Grades 9-11.

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