Question

graph the solution set of the system of inequalities. 2x + y ≤ 4 4x - y ≤ -1 x ≥ -2 y ≤ 7 use the graphing tool to graph the system. click to enlarge graph

Explanation:

Step1: Rewrite inequalities in slope-intercept form

$2x + y \leq 4 \Rightarrow y \leq -2x + 4$
$4x - y \leq -1 \Rightarrow y \geq 4x + 1$
$x \geq -2$ (vertical line), $y \leq 7$ (horizontal line)

Step2: Graph boundary lines

Draw solid lines for all (inequalities have $\leq/\geq$):

• $y = -2x + 4$ (slope -2, y-int 4)
• $y = 4x + 1$ (slope 4, y-int 1)
• $x = -2$ (vertical at $x=-2$)
• $y = 7$ (horizontal at $y=7$)

Step3: Shade solution regions

• Below $y=-2x+4$, above $y=4x+1$, right of $x=-2$, below $y=7$
• Find intersection points of boundaries:
• $x=-2$ & $y=4x+1$: $(-2, -7)$
• $x=-2$ & $y=-2x+4$: $(-2, 8)$ → capped by $y=7$ → $(-2,7)$
• $y=7$ & $y=-2x+4$: $(-1.5,7)$
• $y=-2x+4$ & $y=4x+1$: $(0.5,3)$
• $y=4x+1$ & $y=7$: $(1.5,7)$ → check if satisfies $y \leq -2x+4$? $7 \leq -3 +4=1$ no → so stop at $(0.5,3)$ to $(-1.5,7)$ to $(-2,7)$ to $(-2,-7)$ to $(0.5,3)$

Answer:

The solution set is the polygon bounded by $(-2,-7)$, $(0.5,3)$, $(-1.5,7)$, $(-2,7)$ (shaded region satisfying all inequalities). To graph, plot these points and shade the enclosed area.