which linear inequality is represented by the graph?\n$y\\leq \\frac{1}{2}x + 2$\n$y\\geq \\frac{1}{2}x +…

which linear inequality is represented by the graph?\n$y\\leq \\frac{1}{2}x + 2$\n$y\\geq \\frac{1}{2}x + 2$\n$y\\leq \\frac{1}{3}x + 2$\n$y\\geq \\frac{1}{3}x + 2$

which linear inequality is represented by the graph?\n$y\\leq \\frac{1}{2}x + 2$\n$y\\geq \\frac{1}{2}x + 2$\n$y\\leq \\frac{1}{3}x + 2$\n$y\\geq \\frac{1}{3}x + 2$

Answer

Explanation:

Step1: Find the slope of the line

The line passes through ((0, 2)) and ((4,4)). The slope (m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{4 - 2}{4-0}=\frac{2}{4}=\frac{1}{2}). The (y -)intercept (b = 2), so the equation of the line is (y=\frac{1}{2}x + 2).

Step2: Determine the inequality sign

The line is solid (so the inequality includes equality, (\leq) or (\geq)). Take a test point, say ((0,0)). Substitute into (y) and (\frac{1}{2}x+2): (0) and (\frac{1}{2}(0)+2=2). Since (0\leq2) and ((0,0)) is in the shaded region, the inequality is (y\leq\frac{1}{2}x + 2).

Answer:

(y\leq\frac{1}{2}x + 2) (the first option)