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Answer :
To determine what term can be added to the expression [tex]\(\frac{5}{6} x - 4\)[/tex] to make it equivalent to [tex]\(\frac{1}{2} x - 4\)[/tex], we need to focus on the coefficients of [tex]\(x\)[/tex] in both expressions.
1. Identify the coefficients of [tex]\(x\)[/tex] in each expression:
- In the expression [tex]\(\frac{5}{6} x - 4\)[/tex], the coefficient of [tex]\(x\)[/tex] is [tex]\(\frac{5}{6}\)[/tex].
- In the expression [tex]\(\frac{1}{2} x - 4\)[/tex], the coefficient of [tex]\(x\)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
2. Determine what term needs to be added to [tex]\(\frac{5}{6} x\)[/tex] to obtain [tex]\(\frac{1}{2} x\)[/tex]:
- We are trying to find the term [tex]\(t\)[/tex] such that:
[tex]\[
\frac{5}{6}x + t = \frac{1}{2}x
\][/tex]
3. Solve for [tex]\(t\)[/tex]:
- Subtract [tex]\(\frac{5}{6}x\)[/tex] from both sides of the equation:
[tex]\[
t = \frac{1}{2}x - \frac{5}{6}x
\][/tex]
- Find a common denominator for the fractions:
[tex]\(\frac{1}{2}\)[/tex] is the same as [tex]\(\frac{3}{6}\)[/tex], so:
[tex]\[
t = \frac{3}{6}x - \frac{5}{6}x
\][/tex]
- Simplify the expression:
[tex]\[
t = -\frac{2}{6}x
\][/tex]
- Simplify further:
[tex]\[
t = -\frac{1}{3}x
\][/tex]
Thus, the term [tex]\(-\frac{1}{3}x\)[/tex] can be added to [tex]\(\frac{5}{6} x - 4\)[/tex] to make it equivalent to [tex]\(\frac{1}{2} x - 4\)[/tex]. Therefore, the correct choice is [tex]\(-\frac{1}{3} x\)[/tex].
1. Identify the coefficients of [tex]\(x\)[/tex] in each expression:
- In the expression [tex]\(\frac{5}{6} x - 4\)[/tex], the coefficient of [tex]\(x\)[/tex] is [tex]\(\frac{5}{6}\)[/tex].
- In the expression [tex]\(\frac{1}{2} x - 4\)[/tex], the coefficient of [tex]\(x\)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
2. Determine what term needs to be added to [tex]\(\frac{5}{6} x\)[/tex] to obtain [tex]\(\frac{1}{2} x\)[/tex]:
- We are trying to find the term [tex]\(t\)[/tex] such that:
[tex]\[
\frac{5}{6}x + t = \frac{1}{2}x
\][/tex]
3. Solve for [tex]\(t\)[/tex]:
- Subtract [tex]\(\frac{5}{6}x\)[/tex] from both sides of the equation:
[tex]\[
t = \frac{1}{2}x - \frac{5}{6}x
\][/tex]
- Find a common denominator for the fractions:
[tex]\(\frac{1}{2}\)[/tex] is the same as [tex]\(\frac{3}{6}\)[/tex], so:
[tex]\[
t = \frac{3}{6}x - \frac{5}{6}x
\][/tex]
- Simplify the expression:
[tex]\[
t = -\frac{2}{6}x
\][/tex]
- Simplify further:
[tex]\[
t = -\frac{1}{3}x
\][/tex]
Thus, the term [tex]\(-\frac{1}{3}x\)[/tex] can be added to [tex]\(\frac{5}{6} x - 4\)[/tex] to make it equivalent to [tex]\(\frac{1}{2} x - 4\)[/tex]. Therefore, the correct choice is [tex]\(-\frac{1}{3} x\)[/tex].
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