Question

y >= 4x + 2
y >= \frac{2}{3}x + 2
in which of the following does the shaded region represent the solution set in the xy - plane to the system of inequalities?
choose 1 answer:

Explanation:

Step1: Analyze the y - intercept

The equations of the lines are $y = 4x+2$ and $y=\frac{2}{3}x + 2$. Both lines have a y - intercept of 2, so they cross the y - axis at the point $(0,2)$.

Step2: Analyze the slope and inequality

For the line $y = 4x+2$, the slope $m_1=4$. For the line $y=\frac{2}{3}x + 2$, the slope $m_2=\frac{2}{3}$. Since the inequalities are $y\geq4x + 2$ and $y\geq\frac{2}{3}x+2$, the solution set is above both lines.

Step3: Check the graphs

In option A, the shaded region is above both lines $y = 4x+2$ and $y=\frac{2}{3}x + 2$. In option B, the shaded region is to the left of the lines which does not match the inequalities. In option C, the shaded region is below one of the lines which does not match the inequalities.

Answer:

A.