Select the correct answer.
Identify the expression equivalent to $4(x+x+7)-2 x+8-4$ by substituting $x=1$ and $x=2$.
A. $6 x+11$
B. $3(x+7)$
C. $2(3 x+16)$
D. $3 x+16$
Answer (1)
Simplify the original expression: 4 ( x + x + 7 ) − 2 x + 8 − 4 = 6 x + 32 .
Substitute x = 1 into the simplified expression and the options to find a potential match.
Substitute x = 2 into the simplified expression and the potential match to confirm equivalence.
The equivalent expression is 2 ( 3 x + 16 ) .
Explanation
Understanding the Problem We are given the expression 4 ( x + x + 7 ) − 2 x + 8 − 4 and four possible equivalent expressions: A. 6 x + 11 B. 3 ( x + 7 ) C. 2 ( 3 x + 16 ) D. 3 x + 16
We need to identify the correct equivalent expression by substituting x = 1 and x = 2 into the original expression and the options.
Simplifying the Original Expression First, let's simplify the original expression:
4 ( x + x + 7 ) − 2 x + 8 − 4 = 4 ( 2 x + 7 ) − 2 x + 4 = 8 x + 28 − 2 x + 4 = 6 x + 32
Substituting x=1 into the Original Expression Now, substitute x = 1 into the simplified original expression:
6 ( 1 ) + 32 = 6 + 32 = 38
Substituting x=1 into the Options Next, substitute x = 1 into each of the options and check which one gives 38:
A. 6 ( 1 ) + 11 = 6 + 11 = 17 B. 3 ( 1 + 7 ) = 3 ( 8 ) = 24 C. 2 ( 3 ( 1 ) + 16 ) = 2 ( 3 + 16 ) = 2 ( 19 ) = 38 D. 3 ( 1 ) + 16 = 3 + 16 = 19
Option C is a possible answer.
Substituting x=2 into the Original Expression Now, substitute x = 2 into the simplified original expression:
6 ( 2 ) + 32 = 12 + 32 = 44
Substituting x=2 into Option C Next, substitute x = 2 into option C:
2 ( 3 ( 2 ) + 16 ) = 2 ( 6 + 16 ) = 2 ( 22 ) = 44
Conclusion Since option C matches the original expression for both x = 1 and x = 2 , it is the correct equivalent expression.
Examples
Consider a scenario where you are trying to determine the total cost of items with a discount and an additional fee. Simplifying the expression first makes it easier to calculate the total cost for different quantities of items. For example, if x represents the number of items, the expression 4 ( x + x + 7 ) − 2 x + 8 − 4 could represent the total cost, where 4 ( x + x + 7 ) is the initial cost, − 2 x is a discount based on the number of items, 8 is a fixed fee, and − 4 is a coupon. By simplifying this expression to 2 ( 3 x + 16 ) , you can quickly calculate the total cost for any number of items.
