In the xy-plane, a point (x, y) is in the solution set of the system of inequalities below… — SAT

In the $xy$ -plane, a point $(x, y)$ is in the solution set of the system of inequalities below:

x - 2y \leq -6

x + y \leq 11

y \leq 5

What is the maximum possible value of $x$ ?

Answer: 4

Answer

The maximum possible value of

x

4

The maximum possible value of

x

4

. For all points in the solution set where

y \leq 5

, the active upper bound is

x \leq 2y - 6

. Since

2y - 6

increases as

y

increases, the maximum value of

x

occurs at the maximum boundary

y = 5

, giving

x = 2(5) - 6 = 4

Step-by-Step Solution

Express

x

in terms of

y

using the first two inequalities.

x \leq 2y - 6

and

x \leq 11 - y

To isolate the variable

x

and analyze how its upper bound is constrained by

y

Apply the third inequality constraint,

y \leq 5

, to find the limits on these upper bounds.

2y - 6 \leq 4

and

11 - y \geq 6

Since

y

cannot exceed

5

, the value of

2y - 6

is maximized when

y = 5

, and

11 - y

is minimized when

y = 5

Compare the two upper bounds to determine the active constraint for the domain

y \leq 5

Since

2y - 6 \leq 4

and

11 - y \geq 6

, the active bound is

x \leq 2y - 6

For any point to satisfy the system,

x

must be less than or equal to both bounds, meaning it is restricted by the smaller of the two bounds.

Calculate the maximum value of

x

at the boundary point.

x = 4

at the point

(4, 5)

The function

2y - 6

is increasing with respect to

y

, so its maximum value occurs at the largest possible value of

y

, which is

5

Key Concept

Finding the maximum value of a coordinate within a system of linear inequalities.

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Answer

Step-by-Step Solution

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