Let [tex]x[/tex] be the width of the rectangle, so the length will be [tex]12+x[/tex].
Now, to find the perimeter of our rectangle, we are going to use the formula for the perimeter of a rectangle formula: [tex]P=2(w+l)[/tex]
where
[tex]P[/tex] is the perimeter
[tex]w[/tex] is the width
[tex]l[/tex] is the length
We know that [tex]w=x[/tex] and [tex]l=12+x[/tex], so lets replace those values in our formula:
[tex]P=2(x+12+x)[/tex]
[tex]P=2(2x+12)[/tex]
[tex]P=4x+24[/tex]
We want values of the width that will make the perimeter less than 96 feet, so lets set up our inequality:
[tex]4x+24\ \textless \ 96[/tex]
[tex]4x\ \textless \ 72[/tex]
[tex]x\ \textless \ \frac{72}{4} [/tex]
[tex]x\ \textless \ 18[/tex]
Since the width can't be zero, we can conclude that the values of the width that will make the perimeter less than 96 feet are: [tex]0\ \textless \ x\ \textless \ 18[/tex] or in interval notation: (0,18)
