Answer :
To solve this problem, we want to figure out what term, when added to [tex]\(\frac{5}{6}x - 4\)[/tex], will make it equal to [tex]\(\frac{1}{2}x - 4\)[/tex].
1. Identify the Parts to Compare:
The expressions [tex]\(\frac{5}{6}x - 4\)[/tex] and [tex]\(\frac{1}{2}x - 4\)[/tex] have the same constant term, [tex]\(-4\)[/tex]. Therefore, we only need to focus on the coefficients of [tex]\(x\)[/tex] to make them equivalent.
2. Set the Equation:
We need to find the term that when added to [tex]\(\frac{5}{6}x\)[/tex] results in [tex]\(\frac{1}{2}x\)[/tex].
Let's say this term is [tex]\(t\)[/tex], such that:
[tex]\[
\frac{5}{6}x + t = \frac{1}{2}x
\][/tex]
3. Find the Value of [tex]\(t\)[/tex]:
Start by isolating [tex]\(t\)[/tex]:
[tex]\[
t = \frac{1}{2}x - \frac{5}{6}x
\][/tex]
4. Calculate the Difference:
To subtract these fractions, we need a common denominator. The least common denominator between 2 and 6 is 6. So:
[tex]\[
\frac{1}{2} = \frac{3}{6}
\][/tex]
Thus, the equation becomes:
[tex]\[
t = \frac{3}{6}x - \frac{5}{6}x = \left(\frac{3 - 5}{6}\right)x = \frac{-2}{6}x
\][/tex]
5. Simplify the Result:
Simplify [tex]\(\frac{-2}{6}\)[/tex]:
[tex]\[
\frac{-2}{6} = -\frac{1}{3}
\][/tex]
Hence, the expression for [tex]\(t\)[/tex] is [tex]\(-\frac{1}{3}x\)[/tex].
This tells us that the term we need to add is [tex]\(-\frac{1}{3}x\)[/tex] in order to make the equations equivalent. Therefore, the correct answer is [tex]\(-\frac{1}{3}x\)[/tex].
1. Identify the Parts to Compare:
The expressions [tex]\(\frac{5}{6}x - 4\)[/tex] and [tex]\(\frac{1}{2}x - 4\)[/tex] have the same constant term, [tex]\(-4\)[/tex]. Therefore, we only need to focus on the coefficients of [tex]\(x\)[/tex] to make them equivalent.
2. Set the Equation:
We need to find the term that when added to [tex]\(\frac{5}{6}x\)[/tex] results in [tex]\(\frac{1}{2}x\)[/tex].
Let's say this term is [tex]\(t\)[/tex], such that:
[tex]\[
\frac{5}{6}x + t = \frac{1}{2}x
\][/tex]
3. Find the Value of [tex]\(t\)[/tex]:
Start by isolating [tex]\(t\)[/tex]:
[tex]\[
t = \frac{1}{2}x - \frac{5}{6}x
\][/tex]
4. Calculate the Difference:
To subtract these fractions, we need a common denominator. The least common denominator between 2 and 6 is 6. So:
[tex]\[
\frac{1}{2} = \frac{3}{6}
\][/tex]
Thus, the equation becomes:
[tex]\[
t = \frac{3}{6}x - \frac{5}{6}x = \left(\frac{3 - 5}{6}\right)x = \frac{-2}{6}x
\][/tex]
5. Simplify the Result:
Simplify [tex]\(\frac{-2}{6}\)[/tex]:
[tex]\[
\frac{-2}{6} = -\frac{1}{3}
\][/tex]
Hence, the expression for [tex]\(t\)[/tex] is [tex]\(-\frac{1}{3}x\)[/tex].
This tells us that the term we need to add is [tex]\(-\frac{1}{3}x\)[/tex] in order to make the equations equivalent. Therefore, the correct answer is [tex]\(-\frac{1}{3}x\)[/tex].
