A point (x, y) in the coordinate plane satisfies the system of inequalities below. y \geq … — SAT

A point $(x, y)$ in the coordinate plane satisfies the system of inequalities below.

y \geq 2x - 4

y \leq -x + 5

x \geq 0

y \geq 0

What is the maximum possible value of $x$ for this point?

Answer: 3

Answer

The maximum possible value of

x

that satisfies the system of inequalities is 3.

To find the maximum possible value of

x

for a point satisfying the system of inequalities, we identify the vertices of the region. The upper-right boundary of the region is formed by the intersection of the lines

y = 2x - 4

and

y = -x + 5

. Solving

2x - 4 = -x + 5

gives

3x = 9 \implies x = 3

. At this point,

y = -3 + 5 = 2

, which satisfies the constraints

x \geq 0

and

y \geq 0

. The other boundary vertices of the region are

(0, 0)

(2, 0)

, and

(0, 5)

. Comparing the x-coordinates of these vertices (

0

2

, and

3

), we see that the maximum possible value of

x

3

Step-by-Step Solution

Find the intersection point of the boundary lines

y = 2x - 4

and

y = -x + 5

x = 3

Setting the two boundary line equations equal to each other (

2x - 4 = -x + 5

) allows us to find the x-coordinate where the boundaries cross.

Substitute

x = 3

back into one of the boundary equations to find the y-coordinate.

y = 2

This yields the intersection point

(3, 2)

which lies on both boundary lines.

Verify that the intersection point

(3, 2)

satisfies the other constraints:

x \geq 0

and

y \geq 0

3 \geq 0

and

2 \geq 0

(both true)

The point must lie within the first quadrant to be a valid solution.

Determine the remaining boundary vertices of the solution region in the first quadrant.

Vertices are

(0, 0)

(0, 5)

(2, 0)

, and

(3, 2)

Comparing all vertices will confirm if

(3, 2)

indeed provides the maximum value of

x

Key Concept

Systems of Linear Inequalities in Two Variables

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