Question

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a. (3, -1)
b. (1, 5)
c. (5, -5)
d. (-3, -3)

Explanation:

Step1: Determine the inequality region

The shaded region is above the dashed line (dashed line means the inequality is strict, \(y > mx + b\)). First, find the equation of the line through \((-3, -3)\) and \((3, -1)\).

Slope \(m=\frac{-1 - (-3)}{3 - (-3)}=\frac{2}{6}=\frac{1}{3}\).

Using point - slope form \(y - y_1=m(x - x_1)\) with \((3,-1)\): \(y+1=\frac{1}{3}(x - 3)\), simplify to \(y=\frac{1}{3}x-2\). So the inequality is \(y>\frac{1}{3}x - 2\).

Step2: Test each option

• Option A: \((3,-1)\). Substitute into \(y>\frac{1}{3}x - 2\): \(-1>\frac{1}{3}(3)-2\Rightarrow - 1>1 - 2\Rightarrow - 1>-1\). False, since it's on the line (dashed line, not included).
• Option B: \((1,5)\). Substitute: \(5>\frac{1}{3}(1)-2\Rightarrow5>\frac{1}{3}-2\Rightarrow5>-\frac{5}{3}\). True.
• Option C: \((5,-5)\). Substitute: \(-5>\frac{1}{3}(5)-2\Rightarrow - 5>\frac{5}{3}-2\Rightarrow - 5>-\frac{1}{3}\). False.
• Option D: \((-3,-3)\). Substitute: \(-3>\frac{1}{3}(-3)-2\Rightarrow - 3>-1 - 2\Rightarrow - 3>-3\). False, on the line.

Answer:

B. \((1,5)\)