Question

1. graph the inequality $y > -\frac{1}{2}x + 5$ and $y < x + 2$ on the same coordinate plane. use different colors for each.

a. look at the region that is shaded in both colors. what does this region represent?

b. look at the regions that are shaded in only 1 color. what do these regions represent?

c. look at the region that is not shaded. what does this region represent?

Explanation:

Step1: Graph \( y > -\frac{1}{2}x + 5 \)

• The boundary line is \( y = -\frac{1}{2}x + 5 \), which has a slope of \( -\frac{1}{2} \) and a y - intercept of 5. Since the inequality is \( y > -\frac{1}{2}x + 5 \), the line is dashed (because the inequality is strict, \(>\) not \(\geq\)) and we shade the region above the line.

Step2: Graph \( y < x + 2 \)

• The boundary line is \( y = x + 2 \), which has a slope of 1 and a y - intercept of 2. Since the inequality is \( y < x + 2 \), the line is dashed (because the inequality is strict, \(<\) not \(\leq\)) and we shade the region below the line.

Step3: Analyze part (a)

• The region shaded in both colors is the set of all points \((x,y)\) that satisfy both inequalities \( y > -\frac{1}{2}x + 5 \) and \( y < x + 2 \) simultaneously. In other words, it represents the solution to the system of inequalities \(

$$\begin{cases}y > -\frac{1}{2}x + 5\\y < x + 2\end{cases}$$

\)

Step4: Analyze part (b)

• The regions shaded in only one color represent the set of points that satisfy only one of the two inequalities. For example, the region shaded only for \( y > -\frac{1}{2}x + 5 \) (but not \( y < x + 2 \)) is the set of points that satisfy \( y > -\frac{1}{2}x + 5 \) and \( y\geq x + 2 \) (since it's not in the region of \( y < x + 2 \)), and the region shaded only for \( y < x + 2 \) (but not \( y > -\frac{1}{2}x + 5 \)) is the set of points that satisfy \( y < x + 2 \) and \( y\leq-\frac{1}{2}x + 5 \). In general, these regions represent the solutions to only one of the two inequalities (and do not satisfy the other inequality).

Step5: Analyze part (c)

• The region that is not shaded represents the set of points \((x,y)\) that do not satisfy either \( y > -\frac{1}{2}x + 5 \) or \( y < x + 2 \). In other words, these points satisfy \( y\leq-\frac{1}{2}x + 5 \) and \( y\geq x + 2 \) (the negations of the two original inequalities).

Part (a)

The region shaded in both colors represents the set of all ordered pairs \((x, y)\) that satisfy both inequalities \(y > -\frac{1}{2}x + 5\) and \(y < x + 2\) at the same time. It is the solution to the system of these two linear inequalities.

Regions shaded in only one color represent the set of points that satisfy only one of the two inequalities. For example, a region shaded only for \(y > -\frac{1}{2}x + 5\) has points that satisfy \(y > -\frac{1}{2}x + 5\) but not \(y < x + 2\), and vice - versa. These are solutions to only one of the two inequalities.

The unshaded region consists of points that do not satisfy either \(y > -\frac{1}{2}x + 5\) or \(y < x + 2\). That is, these points satisfy \(y\leq-\frac{1}{2}x + 5\) and \(y\geq x + 2\) (the negations of the two original inequalities).

Answer:

The region represents the solution to the system of inequalities \(\boldsymbol{

$$\begin{cases}y > -\frac{1}{2}x + 5\\y < x + 2\end{cases}$$

}\) (the set of all \((x,y)\) that satisfy both inequalities).

Part (b)