Problem 57 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If the length of a rectangle is 6 inches more than its width, and its perimeter is 24 inches, the distributive property must be used to solve the equation that determines the length.

Step-by-Step Solution

Answer

The statement is false. The distributive property is not needed to solve the given problem.

Step 1: Understand the problem and the statement

First, analyze the statements. We know that the perimeter of a rectangle is given by the formula \(2*(length + width)\). Given that the length is 6 inches more than the width, this relationship can be expressed as \(length = width +6\). We also know that the perimeter is 24 inches.

Step 2: Formulate the Problem Mathematically

Substitute the relationship of length and width into the formula of the perimeter. This leads to \(2*(width + width + 6)= 24\). Simplify that to \(4*width + 12 = 24\).

Step 3: Evaluate the Statement

Solving the resulting equation \(4*width + 12 = 24\) does not require the use of the distributive property. Instead we only need to use subtraction and division to solve for the width. Subtract 12 from both sides to get \(4*width = 12\) then divide by 4 to get \(width = 3\).

Step 4: Conclusion

The statement is false. Instead of the distributive property, subtraction and division are used to solve the equation and determine the width, and consequently, with the relationship between length and width, the length can be determined.