Answer:
[tex]\text{b. } x + 2(x + 2)[/tex]
[tex]\text{The expression } x + 2(x + 2) \text{ gives the same value as } (3x - 2x) + 2(2x - x) + 4 \text{ for both } x = 1 \text{ and } x = 3. \text{ So, the two expressions are equivalent.}[/tex]
[tex]x + 2(x + 2)=(3x - 2x) + 2(2x - x) + 4 \text{ it is true } \forall x \in \mathbb{R}[/tex]
Step-by-step explanation:
[tex](3x - 2x) + 2(2x -x) + 4[/tex]
[tex]x + 4x -2x + 4[/tex]
[tex]3x + 4[/tex]
We are looking for an expression equivalent to [tex]3x + 4[/tex] when [tex]x = 1 \text{ and } x = 3[/tex]
[tex]\text{ for }x=1 : 3(1) + 4=7[/tex]
[tex]\text{ for }x=3 : 3(3) + 4=13[/tex]
Testing all the alternatives:
[tex]\text{ for }x=1 : 2(1+1)-2=2(2)-1=3[/tex] This is not the equivalent expression
[tex]\text{ for }x=1 : x + 2(x + 2) \Rightarrow 1+2(1+2)=1+6=7[/tex]
[tex]\text{ for }x=3 : x + 2(x + 2) \Rightarrow 3+2(3+2)=3+(10)=13[/tex]
This is an equivalent expression
