Intermediate Algebra, Chapter 3, 3.4, Section 3.4, Problem 24

Illustrate the compound inequality $x - y \geq 2$ and $x \geq 3$
Graph the compound inequality $x - y \geq 2 $ and $ x \geq 3$
Since the compound inequality is joined by $and$, then we need to find the intersection of the graphs.
To begin, we graph each of the two inequalities $x - y \geq 2 \text{ and } x \geq 3$ seperately as shown below






Then, we use heavy shading to identify the intersection of the graphs.



To verify this, we choose a test point on the intersection of the region. Let's say point $(4,1)$. So, we have

$
\begin{equation}
\begin{aligned}
x - y &\geq 2 && \text{and} & x &\geq 3\\
\\
4 - 1 &\geq 2 && \text{and} & 4 &\geq 3\\
\\
3 &\geq 2
\end{aligned}
\end{equation}
$


We can see that the ordered pairs we choose inside the intersection of the graph switches both inequalities.