Which pair shows equivalent expressions?
[tex]$3(x+2)=3 x+6$[/tex]
[tex]$3 x+2 x=x^2(3+2)$[/tex]
[tex]$3 x+2=3(x+2)$[/tex]
[tex]$-3(2+x)=-6 x-3$[/tex]
Answer (1)
Distribute the 3 in the first expression of the first pair: 3 ( x + 2 ) = 3 x + 6 .
Simplify the first expression of the second pair: 3 x + 2 x = 5 x and simplify the second expression: x 2 ( 3 + 2 ) = 5 x 2 .
Distribute the 3 in the second expression of the third pair: 3 ( x + 2 ) = 3 x + 6 .
Distribute the -3 in the first expression of the fourth pair: − 3 ( 2 + x ) = − 6 − 3 x .
The first pair shows equivalent expressions: 3 ( x + 2 ) = 3 x + 6 .
Explanation
Understanding the Problem We are given four pairs of expressions and need to determine which pair contains equivalent expressions.
Listing the Pairs The pairs are:
3 ( x + 2 ) and 3 x + 6
3 x + 2 x and x 2 ( 3 + 2 )
3 x + 2 and 3 ( x + 2 )
− 3 ( 2 + x ) and − 6 x − 3
Analyzing Pair 1 Let's analyze each pair:
Pair 1: 3 ( x + 2 ) and 3 x + 6
Distribute the 3 in the first expression: 3 ( x + 2 ) = 3 x + 3 ( 2 ) = 3 x + 6 . This is the same as the second expression, 3 x + 6 . So, this pair shows equivalent expressions.
Analyzing Pair 2 Pair 2: 3 x + 2 x and x 2 ( 3 + 2 )
Simplify the first expression: 3 x + 2 x = 5 x .
Simplify the second expression: x 2 ( 3 + 2 ) = x 2 ( 5 ) = 5 x 2 .
Since 5 x is not equal to 5 x 2 , this pair does not show equivalent expressions.
Analyzing Pair 3 Pair 3: 3 x + 2 and 3 ( x + 2 )
Distribute the 3 in the second expression: 3 ( x + 2 ) = 3 x + 6 .
Since 3 x + 2 is not equal to 3 x + 6 , this pair does not show equivalent expressions.
Analyzing Pair 4 Pair 4: − 3 ( 2 + x ) and − 6 x − 3
Distribute the -3 in the first expression: − 3 ( 2 + x ) = − 3 ( 2 ) + ( − 3 ) ( x ) = − 6 − 3 x .
Since − 6 − 3 x is not equal to − 6 x − 3 , this pair does not show equivalent expressions.
Conclusion Therefore, the only pair that shows equivalent expressions is the first pair: 3 ( x + 2 ) and 3 x + 6 .
Examples
Understanding equivalent expressions is crucial in algebra. For instance, when calculating the area of a rectangular garden with width 'x+2' and a uniform path of width 3 around it, you can express the total width as 3(x+2). Simplifying this to 3x+6 helps in determining the amount of fencing needed, showcasing how algebraic manipulation simplifies real-world problems.
