Question

Find the system of inequalities that encloses the triangular region with vertices: (1,1),(4,1), and (3,5).

Answer 1

To enclose the triangular region with given vertices, we can use a system of inequalities.

Step-by-step explanation:

To find the system of inequalities that encloses the triangular region with vertices (1,1), (4,1), and (3,5), we can use the inequality symbols to show how different metric measurements are related. Let's call the x-coordinate of a point (x, y) as X and the y-coordinate as Y. The system of inequalities can be written as:

1. X >= 1
2. X <= 4
3. Y >= 1
4. Y <= 5
5. X + Y >= 3
6. X - 4Y + 4 <= 0
7. Y - X + 1 >= 0

Answer 2

The system of inequalities that encloses the triangular region with vertices (1,1), (4,1), and (3,5) is y >= 1, y <= 2x - 1, and y <= -4x + 17. These inequalities are derived from the equations of the lines forming the sides of the triangle and are oriented to include the area inside the triangle.

Step-by-step explanation:

To find the system of inequalities that encloses the triangular region with vertices (1,1), (4,1), and (3,5), we need to determine the equations of the lines that make up the sides of the triangle and then decide whether the inequality should be < or > for each equation to enclose the area inside the triangle.

First, let's find the equations of the lines passing through these points:

• Line through (1,1) and (4,1): This horizontal line has the equation y = 1.
• Line through (1,1) and (3,5): To find the slope (m) of this line, we use (y2 - y1) / (x2 - x1). The slope is (5-1) / (3-1) = 4 / 2 = 2. Using point-slope form y - y1 = m(x - x1), we get y - 1 = 2(x - 1), which simplifies to y = 2x - 1.
• Line through (4,1) and (3,5): The slope of this line is (5-1) / (3 - 4) = 4 / -1 = -4. Using the point-slope form again, y - 1 = -4(x - 4), simplifying to y = -4x + 17.

Now, to ensure the inequalities enclose the area inside the triangle, we observe the orientation of the lines and the area with respect to the triangle's vertices:

• For the line y = 1, the region above the line contains the triangle, so the inequality is y >= 1.
• For the line y = 2x - 1, the region below the line contains the triangle, so the inequality is y <= 2x - 1.
• And for the line y = -4x + 17, the region below the line contains the triangle, so the inequality is y <= -4x + 17.

The system of inequalities enclosing the triangle is therefore:

• y >= 1
• y <= 2x - 1
• y <= -4x + 17