Question

Which system of inequalities is represented by the graph?

Answer 1

Answer:

$\left\{\begin{array}{l}y\ge -x-6\\ \\ y\le 2x+4\end{array}\right.$

Step-by-step explanation:

The diagram shows shaded region which boundary is bounded with two lines.

Equation of the 1st line:

This line passes through the points (-6,0) and (0,-6), so its equation is

$\dfrac{x-(-6)}{0-(-6)}=\dfrac{y-0}{-6-0}\\ \\\dfrac{x+6}{6}=\dfrac{y}{-6}\\ \\y=-x-6$

Equation of the 2nd line:

This line passes through the points (-2,0) and (0,4), so its equation is

$\dfrac{x-(-2)}{0-(-2)}=\dfrac{y-0}{4-0}\\ \\\dfrac{x+2}{2}=\dfrac{y}{4}\\ \\y=2x+4$

The origin belongs to the shaded region, so it must satisfy both inequalities, so

$-0-6=-6\le 0\Rightarrow$ correct inequality is $-x-6\le y$

$2\cdot 0+4=4\ge 0\Rightarrow$ correct inequality is $2x+4\ge y$

Note that signsare with notion "or equal to" because lines are solid.

Answer:

$\left\{\begin{array}{l}-x-6\le y\\ \\2x+4\ge y\end{array}\right.\Rightarrow \left\{\begin{array}{l}y\ge -x-6\\ \\ y\le 2x+4\end{array}\right.$