Question

solve the system of two linear inequalities graphically.\

$$\begin{cases}2x + 6y < -12\\\\x \\geq 2\\end{cases}$$

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

Explanation:

Step1: Rewrite first inequality to slope-intercept form

Start with $2x + 6y < -12$. Subtract $2x$ from both sides:
$6y < -2x - 12$
Divide by 6:
$y < -\frac{1}{3}x - 2$
This is a dashed line (since $<$) with slope $-\frac{1}{3}$ and y-intercept $(0, -2)$; shade below the line.

Step2: Graph the second inequality

$x \geq 2$ is a solid vertical line at $x=2$; shade to the right of the line.

Step3: Identify overlapping shaded region

The solution set is the area that is shaded both below the dashed line $y = -\frac{1}{3}x - 2$ and to the right of the solid line $x=2$. There is no overlap between these two regions, so there is no solution.

Answer:

There is no solution (the two shaded regions do not overlap, so no points satisfy both inequalities simultaneously).

To visualize:

1. Draw a dashed line $y = -\frac{1}{3}x - 2$, shade below it.
2. Draw a solid vertical line $x=2$, shade to the right of it.
3. No area is shaded twice, so there is no solution set to graph.