Question

The first line passes through the origin and the point (8, 8). The second line passes through the y-axis at 6 and the point (4, 4). The third line is horizontal through the y-axis at 8. The region to the left of the first line, above the second line, below the horizontal line and right of the y-axis is shaded. What is the minimum value for x over the feasibility region defined by the constraints shown above?

Answer 1

The minimum value of x over the feasibility region defined by the constraints given is 4.

Step-by-step explanation:

The student is asking about the minimum value for x in a feasibility region defined by a set of linear inequalities. The first line passes through the origin (0,0) and the point (8, 8), which implies it has a slope of 1 and an equation of y = x. The second line passes through the y-axis at 6 (0, 6) and the point (4, 4), suggesting it has a negative slope and is represented by the equation y = -0.5x + 6. The third line is horizontal and intersects the y-axis at 8, which means its equation is y = 8. The region described is to the left of the first line (where y > x), above the second line (where y > -0.5x + 6), below the third line (where y < 8), and to the right of the y-axis (where x > 0). The minimum value for x in this region is at the intersection of y = x and y = -0.5x + 6, which is the point where these two lines meet. To find this intersection, we solve the system of equations:

• y = x
• y = -0.5x + 6

By setting the two expressions for y equal to each other:

x = -0.5x + 6

Add 0.5x to both sides:

1.5x = 6

Divide both sides by 1.5:

x = 4

Therefore, the minimum value of x over the feasibility region is 4.