Sure, let's use the distributive property to match equivalent expressions.
The distributive property states that [tex]\( a(b + c) = ab + ac \)[/tex]. This applies similarly for subtraction inside the parentheses. Let's go through each expression and factor them where possible.
### Step-by-Step Breakdown:
1. Expression [tex]\(28 + 7x\)[/tex]:
[tex]\[
28 + 7x = 7 \cdot 4 + 7 \cdot x
\][/tex]
Using the distributive property, we can factor out the common factor of 7:
[tex]\[
28 + 7x = 7(4 + x)
\][/tex]
2. Expression [tex]\(-28 + 7x\)[/tex]:
[tex]\[
-28 + 7x = 7 \cdot (-4) + 7 \cdot x
\][/tex]
Using the distributive property, we can factor out the common factor of 7:
[tex]\[
-28 + 7x = 7(-4 + x)
\][/tex]
3. Expression [tex]\(28 - 7x\)[/tex]:
[tex]\[
28 - 7x = 7 \cdot 4 - 7 \cdot x
\][/tex]
Using the distributive property, we can factor out the common factor of 7:
[tex]\[
28 - 7x = 7(4 - x)
\][/tex]
4. Expression [tex]\(-28 - 7x\)[/tex]:
[tex]\[
-28 - 7x = 7 \cdot (-4) - 7 \cdot x
\][/tex]
Using the distributive property, we can factor out the common factor of 7:
[tex]\[
-28 - 7x = 7(-4 - x)
\][/tex]
### Matching the Equivalent Expressions:
- [tex]\(7(4 + x)\)[/tex] matches [tex]\(28 + 7x\)[/tex]
- [tex]\(7(-4 + x)\)[/tex] matches [tex]\(-28 + 7x\)[/tex]
- [tex]\(7(4 - x)\)[/tex] matches [tex]\(28 - 7x\)[/tex]
- [tex]\(7(-4 - x)\)[/tex] matches [tex]\(-28 - 7x\)[/tex]
Therefore, the equivalent expressions matched using the distributive property are:
- [tex]\(28 + 7x \leftrightarrow 7(4 + x)\)[/tex]
- [tex]\(-28 + 7x \leftrightarrow 7(-4 + x)\)[/tex]
- [tex]\(28 - 7x \leftrightarrow 7(4 - x)\)[/tex]
- [tex]\(-28 - 7x \leftrightarrow 7(-4 - x)\)[/tex]
