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For expressions A-E, select Yes or No to indicate whether each expression is equivalent to [tex]$3(x+2)$[/tex].
A. [tex]$3x + 2$[/tex]
B. [tex][tex]$3(2 + x)$[/tex][/tex] Yes
C. [tex]$3x + 2x$[/tex]
D. [tex]x + 2x + 2 + 4$[/tex]
E. [tex]x + x + x + 1 + 1 + 1 + 1 + 1 + 1$[/tex] No
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
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