Question

the system of inequalities $y \leq -\frac{2}{3}x + 2$ and $y \geq -3x + 1$ is graphed in the $xy$-plane. which of the following is a solution to the system? choose 1 answer: $\boldsymbol{\mathrm{(a)\\ } (-2, 5)}$ $\boldsymbol{\mathrm{(b)\\ } (0, 0)}$ $\boldsymbol{\mathrm{(c)\\ } (3, 1)}$ $\boldsymbol{\mathrm{(d)\\ } (5, -2)}$

Explanation:

Step1: Test Option A (-2,5)

First inequality: $5 \leq -\frac{2}{3}(-2) + 2 = \frac{4}{3} + 2 = \frac{10}{3} \approx 3.33$ → False
Second inequality: $5 \geq -3(-2) + 1 = 6 + 1 = 7$ → False

Step2: Test Option B (0,0)

First inequality: $0 \leq -\frac{2}{3}(0) + 2 = 2$ → True
Second inequality: $0 \geq -3(0) + 1 = 1$ → False

Step3: Test Option C (3,1)

First inequality: $1 \leq -\frac{2}{3}(3) + 2 = -2 + 2 = 0$ → False
Second inequality: $1 \geq -3(3) + 1 = -9 + 1 = -8$ → True

Step4: Test Option D (5,-2)

First inequality: $-2 \leq -\frac{2}{3}(5) + 2 = -\frac{10}{3} + \frac{6}{3} = -\frac{4}{3} \approx -1.33$ → True
Second inequality: $-2 \geq -3(5) + 1 = -15 + 1 = -14$ → True

Answer:

D. (5, -2)