Intermediate Algebra, Chapter 3, 3.4, Section 3.4, Problem 26

Illustrate the compound inequality $3x - y \geq 3$ and $y < 3$

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 $3x - y \geq 3 \text{ and } y < 3$ 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 $(1,-1)$. So, we have

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


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