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Answer :
To find the term that can be added to [tex]\(\frac{5}{6}x - 4\)[/tex] to make it equivalent to [tex]\(\frac{1}{2}x - 4\)[/tex], let's follow these steps:
1. Set up the equation for equivalence:
We want the expressions [tex]\(\frac{5}{6}x - 4\)[/tex] and [tex]\(\frac{1}{2}x - 4\)[/tex] to be equal after adding a certain term to the first expression. Let's call this term [tex]\(T\)[/tex]. Therefore, we have:
[tex]\[
\frac{5}{6}x - 4 + T = \frac{1}{2}x - 4
\][/tex]
2. Cancel out the constants:
Notice that both sides of the equation have [tex]\(-4\)[/tex]. We can eliminate this constant from both sides:
[tex]\[
\frac{5}{6}x + T = \frac{1}{2}x
\][/tex]
3. Isolate the term [tex]\(T\)[/tex]:
To find [tex]\(T\)[/tex], we need to isolate it on one side of the equation. So, subtract [tex]\(\frac{5}{6}x\)[/tex] from both sides:
[tex]\[
T = \frac{1}{2}x - \frac{5}{6}x
\][/tex]
4. Calculate the difference:
Now, we calculate the difference between the coefficients of [tex]\(x\)[/tex]:
[tex]\[
\frac{1}{2} - \frac{5}{6}
\][/tex]
To subtract these fractions, find a common denominator. The least common denominator of 2 and 6 is 6. Convert [tex]\(\frac{1}{2}\)[/tex] to [tex]\(\frac{3}{6}\)[/tex]:
[tex]\[
\frac{3}{6} - \frac{5}{6} = \frac{3 - 5}{6} = \frac{-2}{6} = -\frac{1}{3}
\][/tex]
5. Conclusion:
The term [tex]\(T\)[/tex] that makes the expressions equivalent is [tex]\(-\frac{1}{3}x\)[/tex].
Thus, the correct answer is [tex]\(-\frac{1}{3}x\)[/tex].
1. Set up the equation for equivalence:
We want the expressions [tex]\(\frac{5}{6}x - 4\)[/tex] and [tex]\(\frac{1}{2}x - 4\)[/tex] to be equal after adding a certain term to the first expression. Let's call this term [tex]\(T\)[/tex]. Therefore, we have:
[tex]\[
\frac{5}{6}x - 4 + T = \frac{1}{2}x - 4
\][/tex]
2. Cancel out the constants:
Notice that both sides of the equation have [tex]\(-4\)[/tex]. We can eliminate this constant from both sides:
[tex]\[
\frac{5}{6}x + T = \frac{1}{2}x
\][/tex]
3. Isolate the term [tex]\(T\)[/tex]:
To find [tex]\(T\)[/tex], we need to isolate it on one side of the equation. So, subtract [tex]\(\frac{5}{6}x\)[/tex] from both sides:
[tex]\[
T = \frac{1}{2}x - \frac{5}{6}x
\][/tex]
4. Calculate the difference:
Now, we calculate the difference between the coefficients of [tex]\(x\)[/tex]:
[tex]\[
\frac{1}{2} - \frac{5}{6}
\][/tex]
To subtract these fractions, find a common denominator. The least common denominator of 2 and 6 is 6. Convert [tex]\(\frac{1}{2}\)[/tex] to [tex]\(\frac{3}{6}\)[/tex]:
[tex]\[
\frac{3}{6} - \frac{5}{6} = \frac{3 - 5}{6} = \frac{-2}{6} = -\frac{1}{3}
\][/tex]
5. Conclusion:
The term [tex]\(T\)[/tex] that makes the expressions equivalent is [tex]\(-\frac{1}{3}x\)[/tex].
Thus, the correct answer is [tex]\(-\frac{1}{3}x\)[/tex].
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