Question

which system of linear inequalities is represented by the graph?

• ( y geq \frac{1}{3}x + 3 ) and ( 3x - y > 2 )
• ( y geq \frac{1}{2}x + 3 ) and ( 3x - y > 2 )
• ( y geq \frac{1}{3}x + 3 ) and ( 3x + y > 2 )
• ( y geq \frac{1}{3}x + 3 ) and ( 2x - y > 2 )

Explanation:

Step1: Identify first inequality

First, find the equation of the solid line. It passes through (-3,2) and (0,3). The slope is $\frac{3-2}{0-(-3)}=\frac{1}{3}$, and the y-intercept is 3. Since the shaded region is above the solid line, the inequality is $y\geq \frac{1}{3}x + 3$.

Step2: Identify second inequality

Next, find the equation of the dashed line. It passes through (1,1) and (2,4). The slope is $\frac{4-1}{2-1}=3$. Using point-slope form $y - y_1 = m(x - x_1)$ with (1,1): $y - 1 = 3(x - 1)$, which simplifies to $y = 3x - 2$, or $3x - y = 2$. The shaded region is above this dashed line, so rearrange to get $3x - y > 2$.

Step3: Match to options

The pair of inequalities matches the first option.

Answer:

A. $y\geq \frac{1}{3}x + 3$ and $3x - y > 2$