The system of inequalities is graphed in the -plane. Which point lies in the solution region of the system?

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SAT Linear Inequalities Question 41 | Free SAT Prep | Aniko

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Question 41·Medium·Linear Inequalities in One or Two Variables

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The system of inequalities

y < \tfrac{1}{2}x + 1 \\[0.7em] y \ge -x + 5

is graphed in the $xy$ -plane. Which point $(x, y)$ lies in the solution region of the system?

For SAT questions about systems of linear inequalities with specific answer choices, the fastest approach is usually to plug in each option rather than fully graphing: substitute the $x$ and $y$ values from each choice into each inequality, eliminate any point that makes even one inequality false, and keep going until you find the point that satisfies all inequalities. Pay close attention to strict versus non-strict symbols ( $<$ vs. $\le$ , $>$ vs. $\ge$ ), because equality is allowed only when the inequality includes the line under the symbol.

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Think about what "solution region" means

A point is in the solution region of a system of inequalities only if it satisfies every inequality in the system. If it fails even one inequality, it is not in the solution region.

Start with the first inequality

For each answer choice, plug the $x$ - and $y$ -values into $y < \tfrac12 x + 1$ . If the resulting statement is false, you can eliminate that choice immediately.

Then check the second inequality

Any point that passes the first inequality must also satisfy $y \ge -x + 5$ . Substitute the same $x$ and $y$ into the second inequality and see if the statement is true.

Desmos Guide

Graph the inequalities

In Desmos, type the inequalities exactly as given:

y < 0.5x + 1
y >= -x + 5

Desmos will automatically shade each half-plane; the solution region is where the shadings overlap.

Plot the answer choices as points

Create a table and enter the $x$ - and $y$ -coordinates of the four answer choices in two columns (for example, (2, 2), (4, 2), (1, 5), (0, 4)). Desmos will plot these points on the same graph.

Compare the points to the solution region

Look at each plotted point and see whether it lies inside the overlapping shaded region. The point that lies in this overlapping region is the one that satisfies both inequalities and is therefore the correct choice.

Step-by-step Explanation

Understand what the solution region means

The system of inequalities is

\begin{aligned} y &< \tfrac12 x + 1 \\ y &\ge -x + 5 \end{aligned}

In the $xy$ -plane, each inequality represents a half-plane (one side of a line). The solution region is the set of all points that satisfy both inequalities at the same time. Any answer choice must make both inequalities true when you plug in its $x$ and $y$ values.

Test each point in the first inequality

Use $y < \tfrac12 x + 1$ .

For each choice, substitute $x$ and $y$ :

$(2, 2)$ :
$2 < \tfrac12(2) + 1$ becomes $2 < 1 + 1$ , or $2 < 2$ (this is false).
$(4, 2)$ :
$2 < \tfrac12(4) + 1$ becomes $2 < 2 + 1$ , or $2 < 3$ (this is true).
$(1, 5)$ :
$5 < \tfrac12(1) + 1$ becomes $5 < 0.5 + 1 = 1.5$ (this is false).
$(0, 4)$ :
$4 < \tfrac12(0) + 1$ becomes $4 < 0 + 1$ , or $4 < 1$ (this is false).

Any point that makes this inequality false cannot be in the solution region.

Test points that passed the first inequality in the second inequality

Now use the second inequality $y \ge -x + 5$ on any point(s) that satisfied the first inequality.

Test $(4, 2)$ :

Substitute $x = 4$ , $y = 2$ into $y \ge -x + 5$ :

\begin{aligned} 2 &\ge -4 + 5 \\ 2 &\ge 1 \end{aligned}

This statement is true.

Since $(4, 2)$ makes both $y < \tfrac12 x + 1$ and $y \ge -x + 5$ true, it lies in the solution region of the system. Therefore, the correct answer is $(4, 2)$ .

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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