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egin{aligned} y & ext{ extbackslash leq} \dfrac{ $1$ }{ $3$ }x $-3$ \ y & ext{ extbackslash leq} $-2$ x $-2$ \end{aligned} In which of the following does the shaded region represent the solution set in the $xy$ -plane to the system of inequalities?

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Rational functions: asymptotes and excluded values

Full solution

Q. egin{aligned} y & ext{ extbackslash leq} \dfrac{

1

}{

3

}x

-3

\ y & ext{ extbackslash leq}

-2

x

-2

\end{aligned} In which of the following does the shaded region represent the solution set in the

xy

-plane to the system of inequalities?

Graph Inequality $1$ : The system of inequalities is given by: $\newline$ y \leq \frac{ $1$ }{ $3$ }x - $3$ $\newline$ y \leq $-2$ x - $2$ $\newline$ To find the shaded region that represents the solution set, we need to graph both inequalities on the xy-plane.
Graph Inequality $2$ : First, graph the inequality $y \leq \frac{1}{3}x - 3$ . This is a straight line with a slope of $\frac{1}{3}$ and a $y$ -intercept of $-3$ . The inequality $y \leq \frac{1}{3}x - 3$ means that the solution set includes the area below this line.
Intersection of Shaded Regions: Next, graph the inequality $y \leq -2x - 2$ . This is a straight line with a slope of $-2$ and a $y$ -intercept of $-2$ . The inequality $y \leq -2x - 2$ means that the solution set includes the area below this line as well.
Identify Common Area: The solution set to the system of inequalities is the intersection of the two shaded regions from the previous steps. This is the area that is below both lines on the graph.
Identify Common Area: The solution set to the system of inequalities is the intersection of the two shaded regions from the previous steps. This is the area that is below both lines on the graph.To determine the correct shaded region, we need to look at the graph and identify the area that is common to both inequalities. This is the region that is below both lines.

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