Answer :
To find the term you can add to [tex]\(\frac{5}{6}x - 4\)[/tex] to make it equivalent to [tex]\(\frac{1}{2}x - 4\)[/tex], follow these steps:
### Step 1: Set Up the Equation
Start by setting the expressions equal to each other since we want them to be equivalent after adding the unknown term:
[tex]\[
\frac{5}{6}x - 4 + \text{term} = \frac{1}{2}x - 4
\][/tex]
### Step 2: Eliminate Constant Terms
Since both sides of the equation have [tex]\(-4\)[/tex], subtract [tex]\(-4\)[/tex] from each side to simplify:
[tex]\[
\frac{5}{6}x + \text{term} = \frac{1}{2}x
\][/tex]
### Step 3: Solve for the Term
Now, isolate the term by subtracting [tex]\(\frac{5}{6}x\)[/tex] from both sides:
[tex]\[
\text{term} = \frac{1}{2}x - \frac{5}{6}x
\][/tex]
### Step 4: Simplify the Expression
Convert the fractions to have a common denominator. The common denominator for 2 and 6 is 6. Therefore, rewrite [tex]\(\frac{1}{2}x\)[/tex] as [tex]\(\frac{3}{6}x\)[/tex]:
[tex]\[
\text{term} = \frac{3}{6}x - \frac{5}{6}x = \left( \frac{3 - 5}{6} \right)x = \frac{-2}{6}x
\][/tex]
### Step 5: Simplify Further
Simplify the fraction [tex]\(\frac{-2}{6}\)[/tex]:
[tex]\[
\frac{-2}{6}x = -\frac{1}{3}x
\][/tex]
Therefore, the term you need to add is [tex]\(-\frac{1}{3}x\)[/tex].
The correct choice from the given options is [tex]\(-\frac{1}{3} x\)[/tex].
### Step 1: Set Up the Equation
Start by setting the expressions equal to each other since we want them to be equivalent after adding the unknown term:
[tex]\[
\frac{5}{6}x - 4 + \text{term} = \frac{1}{2}x - 4
\][/tex]
### Step 2: Eliminate Constant Terms
Since both sides of the equation have [tex]\(-4\)[/tex], subtract [tex]\(-4\)[/tex] from each side to simplify:
[tex]\[
\frac{5}{6}x + \text{term} = \frac{1}{2}x
\][/tex]
### Step 3: Solve for the Term
Now, isolate the term by subtracting [tex]\(\frac{5}{6}x\)[/tex] from both sides:
[tex]\[
\text{term} = \frac{1}{2}x - \frac{5}{6}x
\][/tex]
### Step 4: Simplify the Expression
Convert the fractions to have a common denominator. The common denominator for 2 and 6 is 6. Therefore, rewrite [tex]\(\frac{1}{2}x\)[/tex] as [tex]\(\frac{3}{6}x\)[/tex]:
[tex]\[
\text{term} = \frac{3}{6}x - \frac{5}{6}x = \left( \frac{3 - 5}{6} \right)x = \frac{-2}{6}x
\][/tex]
### Step 5: Simplify Further
Simplify the fraction [tex]\(\frac{-2}{6}\)[/tex]:
[tex]\[
\frac{-2}{6}x = -\frac{1}{3}x
\][/tex]
Therefore, the term you need to add is [tex]\(-\frac{1}{3}x\)[/tex].
The correct choice from the given options is [tex]\(-\frac{1}{3} x\)[/tex].
