Question

x + 2y ≥ 2
6x + 2y ≤ -6
a)
image of graph a
b)
image of graph b
c)
image of graph c
d)
image of graph d

Explanation:

Step1: Analyze first inequality \(x + 2y \geq 2\)

Rewrite it as \(y \geq -\frac{1}{2}x + 1\). The boundary line is \(y = -\frac{1}{2}x + 1\) (slope \(-\frac{1}{2}\), y - intercept 1), and we shade above the line (since \(y \geq\)).

Step2: Analyze second inequality \(5x + 2y \leq -6\)

Rewrite it as \(y \leq -\frac{5}{2}x - 3\). The boundary line is \(y = -\frac{5}{2}x - 3\) (slope \(-\frac{5}{2}\), y - intercept - 3), and we shade below the line (since \(y \leq\)).

Step3: Match with graphs

• For \(y \geq -\frac{1}{2}x + 1\), the line has a shallow negative slope and y - intercept 1.
• For \(y \leq -\frac{5}{2}x - 3\), the line has a steeper negative slope and y - intercept - 3.
• The overlapping shaded region should be where both shadings meet. Looking at the options, Option A's shading matches the regions defined by both inequalities (above the first line, below the second line, and the intersection region is correctly shaded).

Answer:

A) (the graph corresponding to option A)