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Look at the system of inequalities. $\newline$ $y \leq -4x + 8$ $\newline$ $x \geq 0$ $\newline$ $y \geq 0$ $\newline$ The solution set is the triangular region where all the inequalities are true. $\newline$ What are the vertices of that triangular region? $\newline$ (,) $\newline$ (,) $\newline$ (,)

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Find the focus or directrix of a parabola

Full solution

Q. Look at the system of inequalities.

\newline

y \leq -4x + 8

\newline

x \geq 0

\newline

y \geq 0

\newline

The solution set is the triangular region where all the inequalities are true.

\newline

What are the vertices of that triangular region?

\newline

(____,____)

\newline

(____,____)

\newline

(____,____)

Find Intersection with X-Axis: First, let's find the intersection of $y = -4x + 8$ and the x-axis ( $y = 0$ ). $\newline$ Set $y$ to $0$ in the equation $y = -4x + 8$ and solve for $x$ . $\newline$ $0 = -4x + 8$ $\newline$ $4x = 8$ $\newline$ $x = 2$ $\newline$ So, the intersection with the x-axis is at $(2, 0)$ .
Find Intersection with Y-Axis: Next, find the intersection of $y = -4x + 8$ and the y-axis ( $x = 0$ ). $\newline$ Set $x$ to $0$ in the equation $y = -4x + 8$ and solve for $y$ . $\newline$ $y = -4(0) + 8$ $\newline$ $y = 8$ $\newline$ So, the intersection with the y-axis is at $(0, 8)$ .
Find Third Vertex: The third vertex is the intersection of $x = 0$ and $y = 0$ , which is the origin. $\newline$ So, the third vertex is at $(0, 0)$ .
Check All Vertices: Now, let's check if we have all the vertices of the triangular region. $\newline$ We have the intersection points of the lines with the axes and the origin, which are all the vertices of the triangular region.

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