Which of the following ordered pairs satisfies both inequalities below? I. II. III. IV.

Solution available with step-by-step explanation

SAT Linear Inequalities Question 73 | Free SAT Prep | Aniko

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Question 73·Easy·Linear Inequalities in One or Two Variables

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Which of the following ordered pairs $(x, y)$ satisfies both inequalities below?

2x + y \le 5

x - y > -1

I. $(0, 5)$
II. $(2, 0)$
III. $(1, 3)$
IV. $(3, 2)$

For questions asking which points satisfy one or more inequalities, the fastest approach is to plug each given point directly into the inequalities rather than solving them algebraically. For each ordered pair, compute the left-hand side of each inequality and quickly check if it meets the condition (remembering the difference between $<$ , $\le$ , $>$ , and $\ge$ ). Discard any point as soon as it fails one inequality, and at the end, match the remaining valid point(s) to the answer choice.

Hints

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Use substitution

For each ordered pair, substitute the x-value and y-value into the expressions $2x + y$ and $x - y$ . Then check whether the resulting numbers make each inequality true.

Be careful with inequality types

Remember that $\le$ allows the left side to be equal to or less than the right side, but $>$ requires the left side to be strictly greater than the right side (equal is not enough).

Stop as soon as a point fails

If an ordered pair fails either inequality, you can immediately discard it and move to the next one—you do not need to check anything else for that point.

Desmos Guide

Graph the inequalities

In Desmos, type the inequalities exactly as given on separate lines:

2x + y <= 5
x - y > -1 Desmos will shade the solution region for each inequality; the overlapping shaded region represents points that satisfy both inequalities.

Plot the candidate points

On new lines, enter each point in curly braces to plot it: {0,5}, {2,0}, {1,3}, and {3,2}. Desmos will display each as a dot on the graph.

Identify which point lies in the overlap

Look for the dot that lies inside the region where the two shadings overlap (the common solution area). The ordered pair corresponding to that dot is the one that satisfies both inequalities; then choose the answer option that lists only that pair.

Step-by-step Explanation

Understand what it means to satisfy both inequalities

We need an ordered pair $(x, y)$ that makes both statements true:

$2x + y \le 5$ ("less than or equal to 5")
$x - y > -1$ ("strictly greater than $-1$ ")

Any pair that fails either one of these inequalities cannot be an answer.

Check point I: (0, 5)

Substitute $x=0$ , $y=5$ into each inequality.

First inequality: $2x + y = 2(0) + 5 = 5$ , and $5 \le 5$ is true.
Second inequality: $x - y = 0 - 5 = -5$ , and $-5 > -1$ is false.

So $(0, 5)$ does not satisfy both inequalities.

Check point II: (2, 0)

Substitute $x=2$ , $y=0$ .

First inequality: $2x + y = 2(2) + 0 = 4$ , and $4 \le 5$ is true.
Second inequality: $x - y = 2 - 0 = 2$ , and $2 > -1$ is also true.

So $(2, 0)$ satisfies both inequalities.

Check points III and IV to confirm they do not work

Now test the remaining points.

Point III: $(1, 3)$

First inequality: $2x + y = 2(1) + 3 = 5$ , and $5 \le 5$ is true.
Second inequality: $x - y = 1 - 3 = -2$ , and $-2 > -1$ is false. So $(1, 3)$ does not satisfy both inequalities.

Point IV: $(3, 2)$

First inequality: $2x + y = 2(3) + 2 = 8$ , and $8 \le 5$ is false. Since it already fails the first inequality, $(3, 2)$ cannot be a solution.

Only one of the four points works.

Match the working point to the answer choice

We found that only point II, $(2, 0)$ , satisfies both inequalities. The answer choice that lists only point II is "II only," which is choice D.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation