Question

provided number line.
A rectangle with a width of 30 cm has a perimeter from 100 cm to 160 cm. Graph a compound inequality that shows the possible lengths of the rectangle.

Answer 1

To find the possible lengths of a rectangle given its width and perimeter constraints, we can start with the formula for the perimeter of a rectangle:

\[
P = 2L + 2W
\]

where \(P\) is the perimeter, \(L\) is the length, and \(W\) is the width. Given that the width \(W\) is 30 cm, we can substitute that into the formula:

\[
P = 2L + 2(30)
\]
\[
P = 2L + 60
\]

Now, we are given that the perimeter \(P\) is between 100 cm and 160 cm. We can write this as a compound inequality:

\[
100 < 2L + 60 < 160
\]

Next, we will solve this compound inequality for \(L\).

1. **Subtract 60 from all parts:**

\[
100 - 60 < 2L < 160 - 60
\]
\[
40 < 2L < 100
\]

2. **Divide all parts by 2:**

\[
20 < L < 50
\]

This gives us the range for the possible lengths of the rectangle.

To graph this compound inequality on a number line:
- Draw a number line.
- Plot open circles (to indicate that 20 and 50 are not included) at 20 and 50.
- Shade the region between 20 and 50.

The final representation of the possible lengths of the rectangle \(L\) would be:

\[
20 < L < 50
\]

This means the length can take any value between 20 cm and 50 cm, but not including 20 cm and 50 cm.