Question

systems of linear inequalities
\

$$\begin{cases} y \\leq x - 3 \\\\ y \\geq -\\frac{1}{2}x - 2 \\end{cases}$$

drag the graph corresponding to the system of inequalities and all points that are part of the solution set into the box.
(3, 0) \quad (-3, -1) \quad (4, -2) \quad (0, 0)

Explanation:

Step1: Analyze \( y \leq x - 3 \)

The line \( y = x - 3 \) has a slope of \( 1 \) and a \( y \)-intercept of \( -3 \). The inequality \( y \leq x - 3 \) means we shade below this line (since \( y \) is less than or equal to the line's value).

Step2: Analyze \( y \geq -\frac{1}{2}x - 2 \)

The line \( y = -\frac{1}{2}x - 2 \) has a slope of \( -\frac{1}{2} \) and a \( y \)-intercept of \( -2 \). The inequality \( y \geq -\frac{1}{2}x - 2 \) means we shade above this line (since \( y \) is greater than or equal to the line's value).

Step3: Find the intersection region

The solution to the system is the region where both shadings overlap. Now, let's check the graphs:

• Graph D: The region where below \( y = x - 3 \) and above \( y = -\frac{1}{2}x - 2 \) overlaps here.

Step4: Check the points

• For \( (3, 0) \): Plug into \( y \leq x - 3 \): \( 0 \leq 3 - 3 = 0 \) (true). Plug into \( y \geq -\frac{1}{2}x - 2 \): \( 0 \geq -\frac{3}{2} - 2 = -\frac{7}{2} \) (true).
• For \( (4, -2) \): Plug into \( y \leq x - 3 \): \( -2 \leq 4 - 3 = 1 \) (true). Plug into \( y \geq -\frac{1}{2}x - 2 \): \( -2 \geq -2 - 2 = -4 \) (true).
• For \( (-3, -1) \): Plug into \( y \leq x - 3 \): \( -1 \leq -3 - 3 = -6 \) (false, so not in solution).
• For \( (0, 0) \): Plug into \( y \leq x - 3 \): \( 0 \leq -3 \) (false, so not in solution).

Answer:

Graph: D
Points: \( (3, 0) \), \( (4, -2) \)