In the xy-plane, a system of inequalities is defined as follows: \begin{aligned} y &\geq 2… — SAT

In the

xy

-plane, a system of inequalities is defined as follows:

\begin{aligned} y &\geq 2x - 4 \\ y &\leq -x + 8 \\ y &\geq \frac{1}{2}x \end{aligned}

How many points

(x, y)

with integer coordinates satisfy this system of inequalities?

Answer: 5

Answer

The total number of points with integer coordinates that satisfy the system is 5.

The system of inequalities defines a closed triangular region in the coordinate plane. Finding the vertices of this triangle gives the horizontal boundaries for

x

, which are

\frac{8}{3} \approx 2.67

and

\frac{16}{3} \approx 5.33

. The only integers in this range are

x = 3

x = 4

, and

x = 5

. Testing each of these integers in the inequalities shows that when

x = 3

y

can be

2, 3, 4,

5

(4 points); when

x = 4

y

can only be

4

(1 point); and when

x = 5

, there are no valid solutions. Adding these together gives a total of 5 points.

Step-by-Step Solution

Find the intersection points of the three boundary lines to define the vertices of the solution region.

The intersection of

y = 2x - 4

and

y = \frac{1}{2}x

is at

x = \frac{8}{3} \approx 2.67, y = \frac{4}{3} \approx 1.33

.
The intersection of

y = -x + 8

and

y = \frac{1}{2}x

is at

x = \frac{16}{3} \approx 5.33, y = \frac{8}{3} \approx 2.67

.
The intersection of

y = 2x - 4

and

y = -x + 8

is at

x = 4, y = 4

Finding the vertices determines the exact boundaries of the solution region in the coordinate plane.

Identify the integer

x

-coordinates that lie within the horizontal span of the region.

The

x

-coordinates of the region range from

\frac{8}{3} \approx 2.67

\frac{16}{3} \approx 5.33

. The integers in this interval are

x = 3

x = 4

, and

x = 5

Since both coordinates must be integers, we only need to test integer values of

x

that fall within the boundaries of the region.

Find the integer

y

-values for each candidate

x

-value.

For

x = 3

: The inequalities require

y \geq 2(3) - 4 = 2

y \leq -3 + 8 = 5

, and

y \geq \frac{1}{2}(3) = 1.5

. Thus,

2 \leq y \leq 5

. The integer solutions are

y = 2, 3, 4, 5

(4 points).
For

x = 4

: The inequalities require

y \geq 2(4) - 4 = 4

y \leq -4 + 8 = 4

, and

y \geq \frac{1}{2}(4) = 2

. Thus,

4 \leq y \leq 4

, which means

y = 4

(1 point).
For

x = 5

: The inequalities require

y \geq 2(5) - 4 = 6

and

y \leq -5 + 8 = 3

. No real number

y

can satisfy both

y \geq 6

and

y \leq 3

(0 points).

Evaluating the system at each candidate

x

-value reveals the set of matching integer

y

-values.

Sum the number of points found for each integer

x

-value.

4 points (when

x = 3

) + 1 point (when

x = 4

) + 0 points (when

x = 5

) = 5 points.

This yields the total number of integer coordinate pairs

(x, y)

that satisfy the system of inequalities.

Key Concept

Analyzing a bounded region defined by a system of linear inequalities to find discrete integer solutions (lattice points).

Alternative Method

Graph the three boundary lines on a grid:

y = 2x - 4

(solid line, shaded above),

y = -x + 8

(solid line, shaded below), and

y = \frac{1}{2}x

(solid line, shaded above). Identify the triangular intersection region on the grid and count the grid intersections (lattice points) that lie within or on the boundaries of this shaded triangle.

Estimated Time:2m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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