Question

Which system of inequalities is represented by the graph?

Answer 1

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

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

`(x-(-6))/(0-(-6))=(y-0)/(-6-0)\\ \\(x+6)/(6)=(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

`(x-(-2))/(0-(-2))=(y-0)/(4-0)\\ \\(x+2)/(2)=(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.`