Exercise 28 Page 279

To find the error in this exercise, we should attempt to verify the graph of the boundary line and the shaded region for each inequality. We have drawn the graph below and added labels to make it easier to describe them.

Boundary Lines

Consider the given system of inequalities. Let's determine whether the boundary lines were graphed correctly.

Inequality (I)	Inequality (II)
From this inequality, we see that the boundary line should have a slope of $3$ and a $y\text{-}$intercept of $4.$ Since the inequality is not strict, the line should be solid.	From this inequality, we can see that the boundary line should have a slope of $\frac{1}{2}$ and a $y\text{-}$intercept of $2.$ Since the inequality is strict, the line should be dashed.

Looking at the graph, we can see that both boundary lines have been graphed correctly.

Shaded Regions

To determine which region to shade, we can use a test point. If we substitute a point into an inequality and it produces a true statement, we should then shade the region containing that point. Otherwise, we shade the opposite region. We will arbitrarily choose to test $(0,0)$ for both inequalities. Let's start by testing $y\le 3x+4.$ Since we obtained a true statement, $(0,0)$ is a solution to this inequality. Therefore, we should shade the region that contains the point. In the given graph, it is showed that this region in fact was correctly shaded. Let's now test the inequality $y>\frac{1}{2}x+2.$ Since it produced a false statement, $(0,0)$ is not a solution to this inequality. Therefore, we should shade the region which does not contain this point. This time, in the given graph they shaded the incorrect region. This is the mistake that was made.

Correct Graph

The correct graph is shown below. We have used the boundary lines and shaded regions as described above.