The length and width of a rectangle are consecutive integers. The perimeter of the rectangle is 86 inches. Find the length and width of the rectangle.

Let's denote the length of the rectangle by \( L \) and the width by \( W \). Since the length and width are consecutive integers, we can express them as:

\[
L = W + 1
\]

The formula for the perimeter \( P \) of a rectangle is given by:

\[
P = 2(L + W)
\]

We know the perimeter is 86 inches:

\[
2(L + W) = 86
\]

Now, substitute \( L = W + 1 \) into the perimeter equation:

\[
2((W + 1) + W) = 86
\]

Simplify the equation:

\[
2(2W + 1) = 86
\]

Divide both sides by 2:

\[
2W + 1 = 43
\]

Subtract 1 from both sides:

\[
2W = 42
\]

Finally, divide by 2:

\[
W = 21
\]

Since \( L = W + 1 \):

\[
L = 21 + 1 = 22
\]

So, the length is 22 inches, and the width is 21 inches.

### Summary:
- **Length:** 22 inches
- **Width:** 21 inches

Would you like a detailed explanation or have any other questions?

Here are five related questions:
1. What is the area of this rectangle?
2. How would the perimeter change if the length and width were both increased by 2 inches?
3. If the width were doubled, what would be the new perimeter?
4. What would the perimeter be if the length and width were both decreased by 3 inches?
5. Can you find another pair of consecutive integers that would give a different perimeter?

**Tip:** Remember, for any two consecutive integers, if you know one, you can always find the other by adding or subtracting 1.

Learn how to find the length and width of a rectangle where the perimeter is 86 inches. Step-by-step solution provided, covering geometric and algebraic concepts. Suitable for Grades 6-8.

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