Which of the following ordered pairs (x, y) is a solution to the system of inequalities be… — SAT

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

\begin{aligned} y &\geq x + 1 \\ y &> -2x + 4 \end{aligned}

$(1, 3)$ Answer
B
$(3, 1)$
C
$(0, 2)$
D
$(-1, 5)$

Answer

The ordered pair

(1, 3)

The ordered pair

(1, 3)

is the correct answer because substituting

x = 1

and

y = 3

into the system of inequalities results in two true statements:

3 \geq 1 + 1

simplifies to

3 \geq 2

(true), and

3 > -2(1) + 4

simplifies to

3 > 2

(true).

Step-by-Step Solution

Substitute the coordinates of the ordered pair

(1, 3)

into the first inequality,

y \geq x + 1

3 \geq 1 + 1 \Rightarrow 3 \geq 2

, which is true.

To be a solution to the system, the ordered pair must satisfy the first inequality.

Substitute the coordinates of the ordered pair

(1, 3)

into the second inequality,

y > -2x + 4

3 > -2(1) + 4 \Rightarrow 3 > 2

, which is true.

The ordered pair must also satisfy the second inequality to be a solution to the system.

Verify that the other ordered pairs do not satisfy both inequalities.

None of the other options satisfy both inequalities:

(3, 1)

fails the first inequality,

(0, 2)

fails the second inequality, and

(-1, 5)

fails the second inequality.

Ensuring that there is exactly one correct solution.

Key Concept

A solution to a system of linear inequalities in two variables is an ordered pair

(x, y)

that makes all inequalities in the system true.

Alternative Method

Alternatively, you can graph the two inequalities on a coordinate plane. The region representing the solution to the system is the overlapping shaded area of both inequalities. By plotting each point, you can visually identify which point lies within this double-shaded region.

Estimated Time:1m 0s

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Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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