Answer:
All real numbers greater than or equal to 2.5 and less than or equal to 5.5
[tex]2.5 \leq x \leq 5.5[/tex]
Step-by-step explanation:
see the attached figure to better understand the problem
step 1
Find the perimeter of the square
The perimeter of the square is
[tex]P=4(b)[/tex]
where
b is the length side of the square
substitute the given value
[tex]P=4(x)=4x\ units[/tex]
step 2
Find the perimeter of rectangle
The perimeter of rectangle is
[tex]P=2(L+W)[/tex]
where
L is the length of rectangle
W is the width of rectangle
substitute the given values
[tex]P=2[(x+1)+3][/tex]
[tex]P=2[x+4][/tex]
[tex]P=(2x+8)\ units[/tex]
step 3
we know that
The difference between the perimeters of the figures is less than or equal to 3
Write an absolute value inequality that represents the situation
[tex]\left|4x-\left(2x+8\right)\right|\le 3[/tex]
[tex]\left|\left(2x-8\right)\right|\le3[/tex]
Solve the absolute value
First case (positive value)
[tex]+(2x-8)\le 3[/tex]
[tex]2x\le 3+8[/tex]
[tex]2x\le 11[/tex]
[tex]x\le 5.5[/tex]
The solution is the interval -----> (-∞,5.5]
Second case (negative value)
[tex]-(2x-8)\le 3[/tex]
Multiply by -1 both sides
[tex](2x-8)\ge -3[/tex]
[tex]2x\ge -3+8[/tex]
[tex]2x\ge 5[/tex]
[tex]x\ge 2.5[/tex]
The solution is the interval -----> [2.5,∞)
The solution of the absolute value for x is
[2.5,∞) ∩ (-∞,5.5] =[2.5,5.5]
[tex]2.5 \leq x \leq 5.5[/tex]
All real numbers greater than or equal to 2.5 and less than or equal to 5.5
