1. Problem 28: Write an algebraic expression for the area of a rectangular rug and find an equivalent expression using properties of operations. The area $A$ of a rectangle is given by the formula: $$A = \text{length} \times \text{width}$$ If the length is $l$ and the width is $w$, then the area is: $$A = l \times w$$ Using properties of operations, such as the distributive property, we can write an equivalent expression if $l$ and $w$ are expressed as sums or differences. For example, if $l = x + 3$ and $w = 2x$, then: $$A = (x + 3)(2x)$$ Using distributive property: $$A = x \times 2x + 3 \times 2x = 2x^2 + 6x$$ This is an equivalent expression for the area. 2. Problem 29: Jamie says $6x - 2x + 4$ and $4(x + 1)$ are not equivalent because one has a subtraction term and the other does not. Let's simplify $6x - 2x + 4$: $$6x - 2x + 4 = (6x - 2x) + 4 = 4x + 4$$ Now expand $4(x + 1)$: $$4(x + 1) = 4x + 4$$ Both expressions simplify to $4x + 4$, so they are equivalent. Jamie's reasoning is incorrect because subtraction inside an expression can be combined and simplified to addition. 3. Problem 31: Chris says $4n - 2$ can be written as $2(2n - 1)$. Let's expand $2(2n - 1)$: $$2(2n - 1) = 2 \times 2n - 2 \times 1 = 4n - 2$$ This matches the original expression exactly, so Chris is correct. Final answers: - Problem 28: Area expression $A = l \times w$ and equivalent expression $A = 2x^2 + 6x$ if $l = x + 3$ and $w = 2x$. - Problem 29: Expressions are equivalent. - Problem 31: Expression can be factored as $2(2n - 1)$.