Sure, let's work through this problem step-by-step to find the missing term in the given expression. The expression provided is:
[tex]\[ --4(12x - 5) + 2x \][/tex]
We need to simplify this expression and find the missing term such that when simplified, the expression becomes [tex]\(-46x + 18\)[/tex].
1. Handle the double negative sign:
The double negative in front of 4 can be simplified. Recall that a negative times a negative is a positive:
[tex]\[ --4(12x - 5) \quad \text{becomes} \quad 4(12x - 5) \][/tex]
2. Distribute the 4 inside the parentheses:
[tex]\[ 4(12x - 5) \][/tex]
Distribute the 4:
[tex]\[ 4 \cdot 12x - 4 \cdot 5 = 48x - 20 \][/tex]
3. Combine the result with the remaining term [tex]\(+2x\)[/tex]:
Now we add the [tex]\(2x\)[/tex] term to our expression:
[tex]\[ 48x - 20 + 2x \][/tex]
Combine like terms:
[tex]\[ 48x + 2x - 20 = 50x - 20 \][/tex]
So, the simplified form of the given expression [tex]\( --4(12x - 5) + 2x \)[/tex] is:
[tex]\[ 50x - 20 \][/tex]
4. Determine the missing term:
We are given that the entire expression simplifies to:
[tex]\[ -46x + 18 \][/tex]
We need this to match our simplified expression [tex]\(50x - 20\)[/tex]. Therefore, we calculate the missing term by determining what, when added to [tex]\(50x - 20\)[/tex], gives us [tex]\(-46x + 18\)[/tex]:
Missing term = [tex]\((-46x + 18) - (50x - 20)\)[/tex]
5. Calculate the missing term:
[tex]\[ (-46x + 18) - (50x - 20) \][/tex]
[tex]\[ = -46x + 18 - 50x + 20 \][/tex]
[tex]\[ = -46x - 50x + 18 + 20 \][/tex]
[tex]\[ = -96x + 38 \][/tex]
So, the missing term is:
[tex]\[ 38 - 96x \][/tex]
This means that the complete original expression must have been:
[tex]\[ (38 - 96x) - 4(12x - 5) + 2x \][/tex]
When simplified, it matches the given simplified form [tex]\(-46x + 18\)[/tex].
