Question

which graph best represents the solution set to the system of inequalities?
$4x + 2y > 8$
$3x - 4y \leq 8$
a
graph a
c
graph c
b
graph b
d
graph d

Explanation:

Step1: Analyze \(4x + 2y>8\)

Rewrite as \(y > - 2x + 4\). The line \(y=-2x + 4\) has a slope of \(-2\) and y - intercept \(4\). Since the inequality is \(>\), the line is dashed, and we shade above the line.

Step2: Analyze \(3x-4y\leq8\)

Rewrite as \(y\geq\frac{3}{4}x - 2\). The line \(y = \frac{3}{4}x-2\) has a slope of \(\frac{3}{4}\) and y - intercept \(-2\). Since the inequality is \(\leq\) (rewritten as \(\geq\) for \(y\)), the line is solid, and we shade above the line.

Step3: Match with graphs

• For \(4x + 2y>8\), the dashed line with slope \(-2\) (going down steeply) and shading above.
• For \(3x - 4y\leq8\), the solid line with slope \(\frac{3}{4}\) (going up gently) and shading above.

Looking at the options, option B has the dashed line for \(4x + 2y>8\) (correct slope and dashed) and the solid line for \(3x - 4y\leq8\) with correct shading.

Answer:

B