Problem 6
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). $$ 2 x-3 y \leq 7 $$
Step-by-Step Solution
Verified Answer
First, we convert the inequality into an equation: \(2x - 3y = 7\). Next, we solve for y: \(y = \frac{2}{3}x - \frac{7}{3}\). We use the origin (0,0) as a test point and determine that the shaded region should include the origin. After sketching the boundary line and shading the region, we find that the region is unbounded and has no corner points.
1Step 1: Analyze the inequality and convert it to an equation
Initially, let's turn the inequality into an equation for easier plotting:
\[
2x - 3y = 7
\]
Now our inequality is:
\[
2x - 3y \leq 7
\]
2Step 2: Determine the boundary line for the inequality
Now let's solve the equation, \(2x - 3y = 7\), for y:
\[
y = \frac{2}{3}x - \frac{7}{3}
\]
This linear equation will help us draw the boundary line for the inequality region.
3Step 3: Determine the region represented by the inequality
To determine which side of the line (above or below) we should shade, we can use a test point. Since the inequality is \(\leq\), if the test point satisfies the inequality, we'll shade the same side as the test point. Conversely, if it doesn't, we'll shade the opposing side.
Let's take the origin (0,0) as the test point. Plugging it into the inequality, we have:
\[
2(0) - 3(0) \leq 7
\]
Since \(0 \leq 7\) is true, we'll shade the region that includes the origin.
4Step 4: Sketch the boundary line and region
Now we will sketch the boundary line (dashed because the inequality includes "equal to") using the equation:
\[
y = \frac{2}{3}x - \frac{7}{3}
\]
We will also shade the region that includes the origin to represent the inequality.
5Step 5: Determine if the region is bounded or unbounded
Looking at our graph, we can see that the shaded region extends infinitely, which means the region is unbounded.
6Step 6: Identify any corner points
Since the region is unbounded, it doesn't have any corner points.
Other exercises in this chapter
Problem 6
\(\begin{array}{ll}\text { Maximize } & p=3 x+2 y+2 z \\ \text { subject to } & x+y+2 z \leq 38 \\ & 2 x+y+z \geq 24 \\ & x \geq 0, y \geq 0, z \geq 0 .\end{arr
View solution Problem 6
\(\begin{array}{ll}\text { Maximize } & p=6 x+y+3 z \\ \text { subject to } & 3 x+y \quad \leq 15 \\ & 2 x+2 y+2 z \leq 20 \\ & x \geq 0, y \geq 0, z \geq 0 .\e
View solution Problem 7
$$ \begin{array}{ll} \text { Minimize } & c=s+3 t+u \\ \text { subject to } & 5 s-t \quad+v \geq 1,000 \\ u-v & \geq 2,000 \\ & \quad s+t \quad \geq 500 \\ & s
View solution Problem 7
Solve the LP problems. If no optimal solution exists, indicate whether the feasible region is empty or the objective function is unbounded. Maximize \(\quad p=3
View solution