Question

study the example showing how to write equivalent expressions. then solve problems 1-9.

example
leroy plants a flower garden that is 4 m long and more than 3 m wide. the expression 4(3 + x) represents the area of the garden. write an expression that is equivalent to 4(3 + x).

you can use tiles to model four groups of 3 + x.

images of tiles representing 3 + x four times, then regrouped as 3+3+3+3 and x+x+x+x
there are (3 + 3 + 3 + 3) + (x + x + x + x) tiles.

1 how do the expressions 4(3 + x) and (3 + 3 + 3 + 3) + (x + x + x + x) compare to one another?

2 simplify the expression 4·3 + 4·x.

3 what property can you apply to write 4(3 + x) as the equivalent expression in problem 2?

4 write two expressions that are equivalent to 2(x + 4).

vocabulary
equivalent expressions
expressions that have the same value
x + 2 + 1 + x and 2x + 3 are equivalent
distributive property
allows you to distribute a factor over the terms in a sum without changing the overall value
2(3 + 7) = 2·3 + 2·7

Explanation:

Step1: Compare expressions

The expression $4(3+x)$ is 4 groups of $(3+x)$, which expands to $(3+3+3+3)+(x+x+x+x)$. Both expressions represent the same total value, so they are equivalent.

Step2: Simplify the expression

Calculate the products:
$4 \cdot 3 = 12$, $4 \cdot x = 4x$
Combine the terms: $12 + 4x$

Step3: Identify the property

The property that distributes a factor over a sum is the distributive property, which transforms $a(b+c)$ to $a\cdot b + a\cdot c$.

Step4: Write equivalent expressions

First, expand using the distributive property: $2\cdot x + 2\cdot 4 = 2x + 8$
Second, write the repeated sum form: $(x+4)+(x+4)$

Answer:

1. They are equivalent expressions (they represent the same value).
2. $12 + 4x$
3. Distributive Property
4. $2x + 8$ and $(x+4)+(x+4)$