Question

the following points are the vertices of a shaded rectangle. (-1,1), (6,1), (6,-3), (-1,-3) a. write a system of linear inequalities represented by the shaded rectangle. left boundary inequality: right boundary inequality: lower boundary inequality: upper boundary inequality: b. find the area of the rectangle. square units

Explanation:

Step1: Determine left - boundary

The left - most x - value of the rectangle is $x=-1$. Points to the right of the line $x = - 1$ are in the rectangle, so the left - boundary inequality is $x\geq - 1$.

Step2: Determine right - boundary

The right - most x - value of the rectangle is $x = 6$. Points to the left of the line $x = 6$ are in the rectangle, so the right - boundary inequality is $x\leq6$.

Step3: Determine lower - boundary

The lower - most y - value of the rectangle is $y=-3$. Points above the line $y=-3$ are in the rectangle, so the lower - boundary inequality is $y\geq - 3$.

Step4: Determine upper - boundary

The upper - most y - value of the rectangle is $y = 1$. Points below the line $y = 1$ are in the rectangle, so the upper - boundary inequality is $y\leq1$.

Step5: Calculate area

The length of the rectangle in the x - direction (base) is $b=6-(-1)=7$. The length of the rectangle in the y - direction (height) is $h = 1-(-3)=4$. The area of a rectangle is $A=b\times h$, so $A=7\times4 = 28$.

Answer:

Left boundary inequality: $x\geq - 1$
Right boundary inequality: $x\leq6$
Lower boundary inequality: $y\geq - 3$
Upper boundary inequality: $y\leq1$
Area of the rectangle: 28 square units