which linear inequality is represented by the graph?\no $y\\leq\\frac{1}{2}x + 2$\no $y\\geq\\frac{1}{2}x +…

which linear inequality is represented by the graph?\no $y\\leq\\frac{1}{2}x + 2$\no $y\\geq\\frac{1}{2}x + 2$\no $y\\leq\\frac{1}{3}x + 2$\no $y\\geq\\frac{1}{3}x + 2$

which linear inequality is represented by the graph?\no $y\\leq\\frac{1}{2}x + 2$\no $y\\geq\\frac{1}{2}x + 2$\no $y\\leq\\frac{1}{3}x + 2$\no $y\\geq\\frac{1}{3}x + 2$

Answer

Explanation:

Step1: Find the slope - intercept form of the line

The equation of a line in slope - intercept form is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. The line passes through the points $(- 4,0)$ and $(0,2)$. The slope $m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{2 - 0}{0-(-4)}=\frac{1}{2}$, and the y - intercept $b = 2$. So the equation of the line is $y=\frac{1}{2}x + 2$.

Step2: Determine the inequality

The line is solid, so the inequality is either $y\leq\frac{1}{2}x + 2$ or $y\geq\frac{1}{2}x + 2$. We test a point in the shaded region, say $(0,0)$. Substitute $x = 0$ and $y = 0$ into the inequalities. For $y\leq\frac{1}{2}x+2$, we have $0\leq\frac{1}{2}(0)+2$, which is $0\leq2$ (true). For $y\geq\frac{1}{2}x + 2$, we have $0\geq\frac{1}{2}(0)+2$, which is $0\geq2$ (false).

Answer:

$y\leq\frac{1}{2}x + 2$