Exercise 27 Page 261

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 one at a time.

Boundary Lines

Let's begin with examining the given system. Inequality (I): From the inequality we see that the boundary line should have a slope of $1$ and a $y\text{-}$intercept of $\text{-}1.$ Since the symbol is $\le ,$ the line should be solid.

Inequality (II): From the inequality we see that the boundary line should have a slope of $1$ and a $y\text{-}$intercept of $3.$ Since the symbol is $\ge ,$ the line should be solid. We have drawn the graph below and added labels to make it easier to describe them.

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

Shaded Regions

To determine which region should be shaded we can use a test point. If we substitute a point into the inequality and it makes a true statement, it is a solution to the inequality. We should then shade the region containing that point. Otherwise we shade the other region. We will arbitrarily choose to test $(0,0)$ for both inequalities, starting with (I). Since $0$ is not less than or equal to $\text{-}1,$ $(0,0)$ is not a solution to this inequality. Thus, we shade the region to the right of the boundary line. However, this is not the case. The given graph shows that the region to the left of the boundary line was shaded. This is a mistake. Now, let's check the region for (II). Since $0$ is not greater than or equal to $3,$ $(0,0)$ is not a solution to this inequality. Thus, we shade the region to the left of the boundary line. However, this is not the case. The given graph shows that the region to the right of the boundary line was shaded. This is another mistake that was made.

Correct Graph

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