Question

consider the system of inequalities and its graph.
$y \geq \frac{x}{4}$
$y \leq x - 3$
(graph: a coordinate plane with two lines and four colored sections labeled 1, 2, 3, 4. the red line is $y = \frac{x}{4}$, the blue line is $y = x - 3$.)
in which section of the graph does the actual solution to the system lie?
\\(\circ\\) 1
\\(\circ\\) 2
\\(\circ\\) 3
\\(\circ\\) 4

Explanation:

Step1: Analyze \( y \geq \frac{x}{4} \)

The inequality \( y \geq \frac{x}{4} \) represents the region above (including) the line \( y = \frac{x}{4} \). From the graph, this is the red - shaded region (upper part relative to \( y=\frac{x}{4} \)).

Step2: Analyze \( y \leq x - 3 \)

The inequality \( y \leq x - 3 \) represents the region below (including) the line \( y=x - 3 \). From the graph, this is the blue - shaded region (lower part relative to \( y = x-3 \)).

Step3: Find the intersection

The solution to the system of inequalities is the intersection of the regions defined by each inequality. We need to find the region that is in both the region of \( y\geq\frac{x}{4} \) and \( y\leq x - 3 \). Looking at the graph, section 3 is the overlapping region of the two shaded areas (the region that is above \( y = \frac{x}{4} \) and below \( y=x - 3 \)).

Answer:

3