\begin{cases} y \geq -2x + 4 \\ y < \frac{1}{2}x + 2 \end{cases} Which of the following or… — SAT

\begin{cases} y \geq -2x + 4 \\ y < \frac{1}{2}x + 2 \end{cases}

Which of the following ordered pairs $(x, y)$ is a solution to the system of inequalities above?

The ordered pair

(1, 2)

is a solution to the system of inequalities.

Substituting the coordinates of the point stating

(1, 2)

into the system of inequalities produces two true statements:

2 \geq -2(1) + 4

simplifies to

2 \geq 2

, which is true; and

2 < \frac{1}{2}(1) + 2

simplifies to

2 < 2.5

, which is also true. Therefore, the ordered pair

(1, 2)

is a solution to the system.

Substitute the coordinates of each option into the first inequality,

y \geq -2x + 4

For

(1, 2)

2 \geq 2

(True). For

(0, -1)

-1 \geq 4

(False). For

(2, 3)

3 \geq 0

(True). For

(0, 3)

3 \geq 4

(False).

To determine which ordered pairs satisfy the first inequality.

Substitute the coordinates of the remaining candidate pairs,

(1, 2)

and

(2, 3)

, into the second inequality,

y < \frac{1}{2}x + 2

For

(1, 2)

2 < 2.5

(True). For

(2, 3)

3 < 3

(False).

To determine which of the remaining candidates also satisfies the second inequality.

Identify the ordered pair that satisfies both inequalities in the system.

Only

(1, 2)

satisfies both inequalities.

An ordered pair is a solution to a system of inequalities if and only if it satisfies all inequalities in the system.

Verifying solutions to a system of linear inequalities by substitution.

Question