To determine which value is possible for the length [tex]\( l \)[/tex] of the rectangle given its perimeter is 16 inches, we need to use the formula for the perimeter of a rectangle:
[tex]\[ 2l + 2w = 16 \][/tex]
We can simplify this equation to find an expression for the width [tex]\( w \)[/tex] in terms of the length [tex]\( l \)[/tex]:
[tex]\[ 2w = 16 - 2l \][/tex]
[tex]\[ w = \frac{16 - 2l}{2} \][/tex]
[tex]\[ w = 8 - l \][/tex]
Now, we need to check each given length to see if it results in a positive width [tex]\( w \)[/tex]:
1. For [tex]\( l = 7 \)[/tex] inches:
[tex]\[ w = 8 - 7 \][/tex]
[tex]\[ w = 1 \][/tex]
Width [tex]\( w \)[/tex] is positive (1 inch), so 7 inches is a possible length.
2. For [tex]\( l = 8 \)[/tex] inches:
[tex]\[ w = 8 - 8 \][/tex]
[tex]\[ w = 0 \][/tex]
Width [tex]\( w \)[/tex] is zero, which typically does not represent a valid rectangle.
3. For [tex]\( l = 9 \)[/tex] inches:
[tex]\[ w = 8 - 9 \][/tex]
[tex]\[ w = -1 \][/tex]
Width [tex]\( w \)[/tex] is negative, which is not possible.
4. For [tex]\( l = 10 \)[/tex] inches:
[tex]\[ w = 8 - 10 \][/tex]
[tex]\[ w = -2 \][/tex]
Width [tex]\( w \)[/tex] is negative, which is not possible.
After checking each possible length, we see that only [tex]\( l = 7 \)[/tex] inches results in a positive width. Therefore, the only possible value for the length of the rectangle is:
[tex]\[ 7 \text{ inches} \][/tex]
