Problem 8
Question
Sketch the region that corresponds to the given inequalities, say whether the region is bounded or unbounded, and find the coordinates of all corner points (if any). $$ y \geq 3 x $$
Step-by-Step Solution
Verified Answer
The region corresponding to the inequality \(y \geq 3x\) is the entire area above the line \(y = 3x\), including the dashed line itself. This region is unbounded, and there are no corner points.
1Step 1: Sketch the boundary line on the coordinate plane.
First, we will sketch the boundary of the inequality. The equation y = 3x represents the boundary line. It passes through the origin (0, 0) and has a slope of 3, which means for every unit increase in x, y will increase by 3 units. Draw the line and make a dashed line since the inequality is "greater than or equal to."
2Step 2: Identify the region satisfying the inequality.
Next, we need to find which region corresponds to the given inequality y >= 3x. Since it is "greater than or equal to," we are looking for the region above the line. This region represents all coordinates (x, y) such that y is greater than or equal to 3x.
3Step 3: Determine if the region is bounded or unbounded.
In this case, there are no other constraints on the x and y values, so there are no boundaries beyond the line y = 3x. Thus, the region is unbounded.
4Step 4: Find the coordinates of all corner points (if any).
Since the region is unbounded, there are no corner points in this problem. The region extends infinitely in the direction above the line y = 3x.
In conclusion, the region corresponding to the inequality y >= 3x is the entire area above the line y = 3x, including the dashed line itself. This region is unbounded, and there are no corner points.
Other exercises in this chapter
Problem 8
Maximize \(\quad p=2 x+2 y+z+2 w\) subject to \(\quad x+y+z+w \leq 50\) \(2 x+y-z-w \geq 10\) \(\quad x+y+z+w \geq 20\) \(x \geq 0, y \geq 0, z \geq 0, w \geq 0
View solution Problem 8
\(\begin{array}{ll}\text { Maximize } & p=3 x+4 y+2 z \\ \text { subject to } & 3 x+y+z \leq 5 \\ & x+2 y+z \leq 5 \\ & x+y+z \leq 4 \\ & x \geq 0, y \geq 0, z
View solution Problem 9
Solve the LP problems. If no optimal solution exists, indicate whether the feasible region is empty or the objective function is unbounded. Minimize \(c=0.2 x+0
View solution Problem 9
\(\begin{aligned} \text { Minimize } & c=6 x+6 y \\ \text { subject to } & x+2 y \geq 20 \\ & 2 x+y \geq 20 \\ & x \geq 0, y \geq 0 . \end{aligned}\)
View solution