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Answer :
To solve this problem, we need to determine what term should be added to the expression [tex]\(\frac{5}{6} x - 4\)[/tex] to make it equivalent to [tex]\(\frac{1}{2} x - 4\)[/tex].
### Step-by-Step Solution:
1. Compare the Expressions:
We want [tex]\(\frac{5}{6} x - 4 + \text{(some term)} = \frac{1}{2} x - 4\)[/tex].
2. Match the Constant Terms:
Notice that the constant terms (-4) on both sides are already the same. So, we only need to focus on matching the coefficients of [tex]\(x\)[/tex].
3. Set Up an Equation for the Coefficients of [tex]\(x\)[/tex]:
Since the constant terms are equal, focus on the [tex]\(x\)[/tex] terms:
[tex]\[
\frac{5}{6} x + \text{(some term)} = \frac{1}{2} x
\][/tex]
This implies:
[tex]\[
\text{some term} = \frac{1}{2} x - \frac{5}{6} x
\][/tex]
4. Calculate the Term:
Subtract the coefficients of [tex]\(x\)[/tex]:
[tex]\[
\frac{1}{2} - \frac{5}{6}
\][/tex]
To subtract these fractions, find a common denominator, which is 6 in this case:
[tex]\[
\frac{1}{2} = \frac{3}{6}
\][/tex]
Then:
[tex]\[
\frac{3}{6} - \frac{5}{6} = -\frac{2}{6}
\][/tex]
Simplifying [tex]\(-\frac{2}{6}\)[/tex], we get:
[tex]\[
-\frac{1}{3}
\][/tex]
5. Conclusion:
The term you need to add is [tex]\(-\frac{1}{3} x\)[/tex]. Therefore, the answer is:
[tex]\[
-\frac{1}{3} x
\][/tex]
This matches the option provided: [tex]\(-\frac{1}{3} x\)[/tex]. Thus, this is the term that will make the two expressions equivalent.
### Step-by-Step Solution:
1. Compare the Expressions:
We want [tex]\(\frac{5}{6} x - 4 + \text{(some term)} = \frac{1}{2} x - 4\)[/tex].
2. Match the Constant Terms:
Notice that the constant terms (-4) on both sides are already the same. So, we only need to focus on matching the coefficients of [tex]\(x\)[/tex].
3. Set Up an Equation for the Coefficients of [tex]\(x\)[/tex]:
Since the constant terms are equal, focus on the [tex]\(x\)[/tex] terms:
[tex]\[
\frac{5}{6} x + \text{(some term)} = \frac{1}{2} x
\][/tex]
This implies:
[tex]\[
\text{some term} = \frac{1}{2} x - \frac{5}{6} x
\][/tex]
4. Calculate the Term:
Subtract the coefficients of [tex]\(x\)[/tex]:
[tex]\[
\frac{1}{2} - \frac{5}{6}
\][/tex]
To subtract these fractions, find a common denominator, which is 6 in this case:
[tex]\[
\frac{1}{2} = \frac{3}{6}
\][/tex]
Then:
[tex]\[
\frac{3}{6} - \frac{5}{6} = -\frac{2}{6}
\][/tex]
Simplifying [tex]\(-\frac{2}{6}\)[/tex], we get:
[tex]\[
-\frac{1}{3}
\][/tex]
5. Conclusion:
The term you need to add is [tex]\(-\frac{1}{3} x\)[/tex]. Therefore, the answer is:
[tex]\[
-\frac{1}{3} x
\][/tex]
This matches the option provided: [tex]\(-\frac{1}{3} x\)[/tex]. Thus, this is the term that will make the two expressions equivalent.
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