Question

solve the system of two linear inequalities graphically. \\(\

$$\begin{cases} y \\leq -x - 2 \\\\ y > 5x + 10 \\end{cases}$$

\\) step 3 of 3: graph the solution set for the system. answer

Explanation:

Step1: Find intersection of lines

Set $-x - 2 = 5x + 10$
$ -x -5x = 10 + 2$
$ -6x = 12$
$ x = -2$
Substitute $x=-2$ into $y=-x-2$: $y = -(-2)-2 = 0$
Intersection point: $(-2, 0)$

Step2: Graph boundary lines

1. For $y \leq -x -2$: Graph solid line $y=-x-2$ (y-intercept $(0,-2)$, slope $-1$). Shade below the line.
2. For $y > 5x +10$: Graph dashed line $y=5x+10$ (y-intercept $(0,10)$, slope $5$). Shade above the line.

Step3: Identify solution region

The solution set is the overlapping shaded area, bounded above by the solid line $y=-x-2$, below by the dashed line $y=5x+10$, and ending at the intersection point $(-2, 0)$.

Answer:

The solution set is the region that is:

• Below and including the solid line $y = -x - 2$
• Above (but not including) the dashed line $y = 5x + 10$
• Bounded on the right by the intersection point $(-2, 0)$

(Graphically, this is a wedge-shaped region opening to the left, with the vertex at $(-2, 0)$)