Question

text description for graph
a. $y \leq -4x + 2$
b. $y \geq 4x + 2$
c. $y \leq 4x + 2$
d. $y \geq -4x + 2$

Explanation:

Step1: Find the slope of the line

The slope \( m \) between two points \((x_1,y_1)=(0,2)\) and \((x_2,y_2)=(-1,-2)\) is calculated using the formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \).
Substituting the values, we get \( m=\frac{-2 - 2}{-1 - 0}=\frac{-4}{-1} = 4 \).

Step2: Determine the equation of the line

Using the slope-intercept form \( y = mx + b \), where \( m = 4 \) and \( b = 2 \) (from the y-intercept \((0,2)\)), the equation of the line is \( y=4x + 2 \).

Step3: Determine the inequality

The shaded region is above the line, and the line is solid (so the inequality includes equality). We test a point in the shaded region, say \((0,3)\) (which is in the shaded area).
For option B: \( 3\geq4(0)+2\) simplifies to \( 3\geq2 \), which is true.
For option C: \( 3\leq4(0)+2\) simplifies to \( 3\leq2 \), which is false.
For option A: \( 3\leq - 4(0)+2\) simplifies to \( 3\leq2 \), false.
For option D: \( 3\geq - 4(0)+2\) simplifies to \( 3\geq2 \), but the slope we found is 4, not - 4, so D is incorrect.

Answer:

B. \( y\geq4x + 2 \)