Question

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persevere graph the system of inequalities $y \geq 1$, $y \leq x + 4$, and $y \leq -x + 4$ on a separate sheet of paper. estimate the area that represents the solution.
____ units$^2$

Explanation:

Step1: Find intersection points

First, find where the boundary lines intersect:

1. Intersection of $y = 1$ and $y = x + 4$:

Set $1 = x + 4$, so $x = -3$. Point: $(-3, 1)$

1. Intersection of $y = 1$ and $y = -x + 4$:

Set $1 = -x + 4$, so $x = 3$. Point: $(3, 1)$

1. Intersection of $y = x + 4$ and $y = -x + 4$:

Set $x + 4 = -x + 4$, so $2x=0$, $x=0$, $y=4$. Point: $(0, 4)$

Step2: Identify shape & dimensions

The solution region is a triangle.
Base length: Distance between $(-3,1)$ and $(3,1)$ is $3 - (-3) = 6$ units.
Height: Vertical distance from $(0,4)$ to the base $y=1$ is $4 - 1 = 3$ units.

Step3: Calculate triangle area

Use area formula for triangle: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$
$\text{Area} = \frac{1}{2} \times 6 \times 3$

Answer:

9 units$^2$