Answer:
The width, x, is in the interval (0, 7), or 0 < x < 7.
Explanation:
width = x
length = 7x + 7
area = x(7x + 7)
x(7x + 7) < 392
7x² + 7x - 392 < 0
x² + x - 56 < 0
(x - 7)(x + 8) < 0
We have two points of interest on the number line:
-8 and 7
That makes three intervals of interest:
(-∞, -8), (-8, 7), (7, ∞)
Now we test a number from each interval.
Try -10:
(x - 7)(x + 8) < 0
(-10 - 7)(-10 + 8) < 0
(-17)(-2) < 0
34 < 0 False
Interval (-∞, -8) does not work.
Try 0:
(x - 7)(x + 8) < 0
(0 - 7)(0 + 8) < 0
(-7)(8) < 0
-56 < 0 True
Interval (-8, 7) works.
Try 10:
(x - 7)(x + 8) < 0
(10 - 7)(10 + 8) < 0
(3)(18) < 0
54 < 0 False
Interval (7, ∞) does not work.
Since we are dealing with a rectangle, the length and width must be positive numbers.
The interval (-8, 7) works mathematically for the width, but it must be limited to positive numbers, so the width must be in the interval
(0, 7)