Question

select the correct answer from each drop - down menu.
a system of two linear inequalities is graphed as shown, where the solution region is shaded.
complete the sentences below by determining whether each point is, or is not, located in the solution region.
the point (-3,2)
the point (-7,5)
the point (2,-5)
the point (-14,6)
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Explanation:

Step1: Find inequalities from graph

First, identify the two lines:

1. Solid line: passes through $(-6,0)$ and $(0,-2)$. Slope $m=\frac{-2-0}{0-(-6)}=-\frac{1}{3}$. Equation: $y=-\frac{1}{3}x-2$. Shading is above this line, so inequality: $y\geq -\frac{1}{3}x-2$ (or $x+3y\geq -6$).
2. Dashed line: passes through $(0,4)$ and $(3,2)$. Slope $m=\frac{2-4}{3-0}=-\frac{2}{3}$. Equation: $y=-\frac{2}{3}x+4$. Shading is below this line, so inequality: $y< -\frac{2}{3}x+4$ (or $2x+3y<12$).

Step2: Test point (-3,2)

Substitute into both inequalities:

1. $2\geq -\frac{1}{3}(-3)-2 \implies 2\geq 1-2 \implies 2\geq -1$ (True)
2. $2< -\frac{2}{3}(-3)+4 \implies 2< 2+4 \implies 2<6$ (True)

Both are true, so the point is in the solution region.

Step3: Test point (-7,5)

Substitute into both inequalities:

1. $5\geq -\frac{1}{3}(-7)-2 \implies 5\geq \frac{7}{3}-2 \implies 5\geq \frac{1}{3}$ (True)
2. $5< -\frac{2}{3}(-7)+4 \implies 5< \frac{14}{3}+4 \implies 5< \frac{26}{3}\approx8.67$ (True)

Both are true, so the point is in the solution region.

Step4: Test point (2,-5)

Substitute into both inequalities:

1. $-5\geq -\frac{1}{3}(2)-2 \implies -5\geq -\frac{2}{3}-2 \implies -5\geq -\frac{8}{3}\approx-2.67$ (False)

One inequality fails, so the point is NOT in the solution region.

Step5: Test point (-14,6)

Substitute into both inequalities:

1. $6\geq -\frac{1}{3}(-14)-2 \implies 6\geq \frac{14}{3}-2 \implies 6\geq \frac{8}{3}\approx2.67$ (True)
2. $6< -\frac{2}{3}(-14)+4 \implies 6< \frac{28}{3}+4 \implies 6< \frac{40}{3}\approx13.33$ (True)

Both are true, so the point is in the solution region.

Answer:

The point (-3,2) $\boldsymbol{\checkmark}$ located in the solution region.
The point (-7,5) $\boldsymbol{\checkmark}$ located in the solution region.
The point (2,-5) $\boldsymbol{\square}$ located in the solution region.
The point (-14,6) $\boldsymbol{\checkmark}$ located in the solution region.