Intermediate Algebra, Chapter 3, 3.4, Section 3.4, Problem 28

Illustrate the compound inequality $6x - 4y < 10$ and $y > 2$

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 $6x - 4y < 10 \text{ and } y > 2$ 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 $(1,3)$. So, we have

$
\begin{equation}
\begin{aligned}
6x - 4y &< 10 && \text{and} & y &> 2 \\
\\
6(1) - 4(3) &< 10 && \text{and} & 3 &> 2\\
\\
6 - 12 &< 10 \\
\\
-6 &< 10
\end{aligned}
\end{equation}
$


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