Question

which graph best represents the solution set to the system of inequalities?
$2x + 3y > -12$
$y \geq 3x - 1$
a
graph a
c
graph c
b
graph b
d
graph d

Explanation:

Step1: Analyze \(2x + 3y > -12\)

Rewrite as \(y > -\frac{2}{3}x - 4\). The boundary line \(y = -\frac{2}{3}x - 4\) has a slope of \(-\frac{2}{3}\) and y - intercept \(-4\). Since the inequality is \(>\), the line is dashed. Test \((0,0)\): \(0 > - 4\), so shade above the dashed line.

Step2: Analyze \(y \geq 3x - 1\)

The boundary line \(y = 3x - 1\) has a slope of \(3\) and y - intercept \(-1\). Since the inequality is \(\geq\), the line is solid. Test \((0,0)\): \(0\geq - 1\), so shade above the solid line.

Step3: Match with graphs

• Graph A: Solid line for \(2x + 3y > - 12\) (wrong, should be dashed).
• Graph B: Solid line for \(2x + 3y > - 12\) (wrong, should be dashed).
• Graph C: Dashed line for \(2x + 3y > - 12\) (correct line, slope and intercept), solid line for \(y\geq3x - 1\) (correct line, slope and intercept), and shading above both (matches the test points).
• Graph D: Shading below \(y\geq3x - 1\) (wrong, should be above).

Answer:

C