If the system of inequalities $\newline$ $y \leq -3x-4$ and $\newline$ $y \leq 3x+8$ is graphed in the $\newline$ $xy$ -plane, which quadrant contains no solutions to the system? $\newline$ Choose $1$ answer: $\newline$ (A) Quadrant I $\newline$ (B) Quadrant II $\newline$ (C) Quadrant III $\newline$ (D) Quadrant IV

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Q. If the system of inequalities

\newline

y \leq -3x-4

and

\newline

y \leq 3x+8

is graphed in the

\newline

xy

-plane, which quadrant contains no solutions to the system?

\newline

Choose

1

answer:

\newline

(A) Quadrant I

\newline

(B) Quadrant II

\newline

\newline

(D) Quadrant IV

Understand Inequalities: First, let's understand the inequalities given. The first inequality is $y \leq -3x - 4$ , and the second inequality is $y \leq 3x + 8$ . These inequalities define regions in the $xy$ -plane where their respective conditions are met. To find out which quadrant contains no solutions to the system, we need to analyze the direction in which these inequalities point on the graph.
Graph First Inequality: Graph the first inequality $y \leq -3x - 4$ . This line has a negative slope and crosses the y-axis at $-4$ . The inequality $y \leq -3x - 4$ means that the solution set includes the area below this line. This area will cover parts of Quadrants $III$ , $IV$ , and $II$ .
Graph Second Inequality: Graph the second inequality $y \leq 3x + 8$ . This line has a positive slope and crosses the $y$ -axis at $8$ . The inequality $y \leq 3x + 8$ means that the solution set includes the area below this line. This area will cover parts of Quadrants $III$ , $IV$ , and $I$ .
Identify Quadrant Solutions: To find the quadrant with no solutions to the system, we need to identify where the solution sets of the two inequalities do not overlap. From the previous steps, we know that the solution set of the first inequality covers parts of Quadrants $ext{III}$ , $ext{IV}$ , and $ext{II}$ , and the solution set of the second inequality covers parts of Quadrants $ext{III}$ , $ext{IV}$ , and $ext{I}$ .
Consider Overlapping Sets: Considering the overlap of the solution sets from both inequalities, we can see that Quadrants $ext{III}$ and $ext{IV}$ are common to both, indicating that they contain solutions. Quadrant $ext{II}$ is only mentioned in the context of the first inequality, and Quadrant $ext{I}$ is only mentioned in the context of the second inequality. However, since both lines have areas below them that extend into Quadrants $ext{I}$ and $ext{II}$ , we need to consider the direction of the inequalities more carefully.
Find Quadrant with No Solutions: Upon closer examination, we realize that the intersection of the solution sets (areas below both lines) does not include Quadrant I. This is because Quadrant I is above both lines for positive values of $x$ and $y$ , which do not satisfy either of the inequalities when considered together. Therefore, Quadrant I is the quadrant that contains no solutions to the system of inequalities.