The linear inequality given is:

$x - y \geq 3$

Let's solve and graph this step by step:

Step 1: Rewrite the inequality

We first rearrange the inequality to express $y$ in terms of $x$:

$x - y \geq 3$

Subtract $x$ from both sides:

$-y \geq 3 - x$

Now, multiply through by $-1$ to solve for $y$, remembering to reverse the inequality:

$y \leq x - 3$

Step 2: Graph the boundary line

The boundary line is the equation $y = x - 3$. This line has a slope of 1 and a y-intercept of $-3$. Since the inequality is $y \leq x - 3$, the graph will include the area below this line.

• Solid Line: Because the inequality is $\leq$, we will draw a solid line to include points on the line $y = x - 3$.
• Shaded Region: The region below the line represents the solutions to the inequality $y \leq x - 3$.

Step 3: Shading the region

Since the inequality is $y \leq x - 3$, we will shade the area below the line $y = x - 3$.

I can now visualize this inequality graph for you.

Should I generate the graph for this inequality?

Related Questions:

1. How do we graph inequalities with different slopes?
2. What is the effect of switching the inequality symbol on the graph?
3. How can you check whether a point lies in the solution set of an inequality?
4. What changes occur if we use $y \geq x - 3$ instead of $y \leq x - 3$?
5. How does the slope-intercept form help in graphing linear inequalities?

Tip:

Always check which side of the boundary line you need to shade by testing a point, such as $(0, 0)$, in the inequality.