Let's compare each given algebraic expression to the expression [tex]\( 3(x + 2) \)[/tex] to determine if they are equivalent.
Consider [tex]\( 3(x + 2) \)[/tex]:
[tex]\[ 3(x + 2) = 3 \cdot x + 3 \cdot 2 = 3x + 6 \][/tex]
Now, we'll analyze each expression:
Expression A: [tex]\( 3x + 2 \)[/tex]
[tex]\[ 3x + 2 \][/tex] does not simplify further to [tex]\( 3x + 6 \)[/tex]. The term [tex]\( +2 \)[/tex] is not the same as [tex]\( +6 \)[/tex], so this is not equivalent.
[tex]\[ \boxed{\text{No}} \][/tex]
Expression B: [tex]\( 3(2 + x) \)[/tex]
Using the distributive property:
[tex]\[ 3(2 + x) = 3 \cdot 2 + 3 \cdot x = 6 + 3x = 3x + 6 \][/tex]
This matches [tex]\( 3(x + 2) \)[/tex], so this is equivalent.
[tex]\[ \boxed{\text{Yes}} \][/tex]
Expression C: [tex]\( 3x + 2x \)[/tex]
Combine like terms:
[tex]\[ 3x + 2x = 5x \][/tex]
[tex]\( 5x \)[/tex] does not match [tex]\( 3x + 6 \)[/tex], so this is not equivalent.
[tex]\[ \boxed{\text{No}} \][/tex]
Expression D: [tex]\( x + 2x + 2 + 4 \)[/tex]
Combine like terms:
[tex]\[ x + 2x = 3x \][/tex]
And combine the constants:
[tex]\[ 2 + 4 = 6 \][/tex]
So [tex]\( x + 2x + 2 + 4 = 3x + 6 \)[/tex]
This matches [tex]\( 3(x + 2) \)[/tex], so this is equivalent.
[tex]\[ \boxed{\text{Yes}} \][/tex]
Expression E: [tex]\( x + x + x + 1 + 1 + 1 + 1 + 1 + 1 \)[/tex]
Combine like terms:
[tex]\[ x + x + x = 3x \][/tex]
And combine the constants:
[tex]\[ 1 + 1 + 1 + 1 + 1 + 1 = 6 \][/tex]
So [tex]\( x + x + x + 1 + 1 + 1 + 1 + 1 + 1 = 3x + 6 \)[/tex]
This matches [tex]\( 3(x + 2) \)[/tex], so this is equivalent.
[tex]\[ \boxed{\text{Yes}} \][/tex]
Therefore, the equivalency for each expression compared to [tex]\( 3(x + 2) \)[/tex] is:
A [tex]\( 3x + 2 \)[/tex] - No
B [tex]\( 3(2 + x) \)[/tex] - Yes
C [tex]\( 3x + 2x \)[/tex] - No
D [tex]\( x + 2x + 2 + 4 \)[/tex] - Yes
E [tex]\( x + x + x + 1 + 1 + 1 + 1 + 1 + 1 \)[/tex] - Yes
