The inequality given is $x - y \geq 5$. To graph this inequality on the coordinate plane, we need to follow a few steps:

Step 1: Convert the inequality to an equation.

Convert the inequality $x - y \geq 5$ into the equation: $x - y = 5$ This is a straight line.

Step 2: Rearrange the equation into slope-intercept form.

Rearranging $x - y = 5$ gives: $y = x - 5$ This is a linear equation where the slope is 1 and the y-intercept is -5.

Step 3: Plot the boundary line.

• The y-intercept is at $(0, -5)$, so plot that point.
• Since the slope is 1, for every increase of 1 unit in $x$, $y$ increases by 1 unit. So another point is $(1, -4)$.
• Continue plotting points and draw the line.

Since the inequality is $\geq$, the line itself is included in the solution, meaning it should be solid.

Step 4: Shade the appropriate region.

Now, we need to determine which side of the line to shade. Since the inequality is $x - y \geq 5$, you shade the region above the line $y = x - 5$, because that's where the values of $x - y$ are greater than or equal to 5.

Summary:

• The boundary line is $y = x - 5$.
• The region above the line is shaded.
• The line itself is solid since the inequality includes "equal to" ($\geq$).

Would you like further clarification on this graphing process or any additional details?

Here are 5 follow-up questions:

1. How does changing the slope affect the graph of the inequality?
2. What happens if the inequality was $x - y < 5$ instead?
3. How do you determine where to shade for different types of inequalities?
4. What is the difference between strict inequalities ($<, >$) and non-strict inequalities ($\leq, \geq$) in graphing?
5. Can inequalities involving absolute values be graphed similarly?

Tip: Always test a point, such as $(0,0)$, to verify which side of the boundary line satisfies the inequality.