Question

in exercises 3–6, tell whether the ordered pair is a solution of the system of linear inequalities.

1. (-4, 3)
2. (-3, -1)
3. (-2, 5)
4. (1, 1)

Explanation:

First, we derive the system of inequalities from the graph:

1. The solid line has a slope of $1$ and y-intercept of $3$, so its equation is $y = x + 3$. The shaded region is below this line, so the inequality is $y \leq x + 3$.
2. The dashed line has a slope of $-1$ and x-intercept of $-2$, so its equation is $y = -x - 2$. The shaded region is below this line, so the inequality is $y < -x - 2$.

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Exercise 3: $(-4, 3)$

Step1: Test $y \leq x + 3$

Substitute $x=-4, y=3$:
$3 \leq -4 + 3$
$3 \leq -1$ (False)

Step2: Test $y < -x - 2$

Substitute $x=-4, y=3$:
$3 < -(-4) - 2$
$3 < 2$ (False)

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Exercise 4: $(-3, -1)$

Step1: Test $y \leq x + 3$

Substitute $x=-3, y=-1$:
$-1 \leq -3 + 3$
$-1 \leq 0$ (True)

Step2: Test $y < -x - 2$

Substitute $x=-3, y=-1$:
$-1 < -(-3) - 2$
$-1 < 1$ (True)

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Exercise 5: $(-2, 5)$

Step1: Test $y \leq x + 3$

Substitute $x=-2, y=5$:
$5 \leq -2 + 3$
$5 \leq 1$ (False)

Step2: Test $y < -x - 2$

Substitute $x=-2, y=5$:
$5 < -(-2) - 2$
$5 < 0$ (False)

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Exercise 6: $(1, 1)$

Step1: Test $y \leq x + 3$

Substitute $x=1, y=1$:
$1 \leq 1 + 3$
$1 \leq 4$ (True)

Step2: Test $y < -x - 2$

Substitute $x=1, y=1$:
$1 < -(1) - 2$
$1 < -3$ (False)

Answer:

1. No, $(-4, 3)$ is not a solution.
2. Yes, $(-3, -1)$ is a solution.
3. No, $(-2, 5)$ is not a solution.
4. No, $(1, 1)$ is not a solution.