Question

1. which graph shows the solution to the system of inequalities

$3x - 2y > 2$
$2x + 3y \leq 18$
(there are four graphs labeled a, b, c, d with corresponding radio buttons)

Explanation:

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

Rewrite it in slope - intercept form (\(y=mx + b\)):
\(-2y>-3x + 2\)
Divide both sides by \(-2\) (remember to reverse the inequality sign):
\(y<\frac{3}{2}x - 1\)
The boundary line \(y = \frac{3}{2}x-1\) has a slope of \(\frac{3}{2}\) and a \(y\) - intercept of \(- 1\). Since the inequality is \(y<\frac{3}{2}x - 1\), the line is dashed and we shade below the line.

Step2: Analyze \(2x + 3y\leq18\)

Rewrite it in slope - intercept form:
\(3y\leq - 2x + 18\)
\(y\leq-\frac{2}{3}x + 6\)
The boundary line \(y=-\frac{2}{3}x + 6\) has a slope of \(-\frac{2}{3}\) and a \(y\) - intercept of \(6\). Since the inequality is \(y\leq-\frac{2}{3}x + 6\), the line is solid and we shade below the line.

Step3: Analyze the intersection and the graphs

• For the inequality \(y<\frac{3}{2}x - 1\), the region is below the dashed line with slope \(\frac{3}{2}\).
• For the inequality \(y\leq-\frac{2}{3}x + 6\), the region is below the solid line with slope \(-\frac{2}{3}\).

Now, let's analyze the options:

• Option A: The regions do not match the correct shading for both inequalities.
• Option B: The blue region (for \(y\leq-\frac{2}{3}x + 6\)) and the red region (for \(y<\frac{3}{2}x - 1\)) and their intersection (the purple region) match the correct shading for both inequalities. The line \(y = \frac{3}{2}x-1\) is dashed, and \(y=-\frac{2}{3}x + 6\) is solid, and the shading is correct.
• Option C: The shading regions do not match the required regions for the two inequalities.
• Option D: The shading regions do not match the required regions for the two inequalities.

Answer:

B