Answer :
We start with the expression
[tex]$$\frac{5}{6}x - 4$$[/tex]
and we want to determine the term that, when added, makes it equivalent to
[tex]$$\frac{1}{2}x - 4.$$[/tex]
Since both expressions have the same constant [tex]$-4$[/tex], the adjustment must be made only to the [tex]$x$[/tex]-term.
Let the term we add be [tex]$t$[/tex]. This means we need:
[tex]$$\frac{5}{6}x + t = \frac{1}{2}x.$$[/tex]
To find [tex]$t$[/tex], subtract [tex]$\frac{5}{6}x$[/tex] from both sides:
[tex]$$
t = \frac{1}{2}x - \frac{5}{6}x.
$$[/tex]
To subtract these fractions, express [tex]$\frac{1}{2}$[/tex] with denominator 6:
[tex]$$
\frac{1}{2} = \frac{3}{6}.
$$[/tex]
Thus, we have:
[tex]$$
t = \frac{3}{6}x - \frac{5}{6}x = \left(\frac{3-5}{6}\right)x = -\frac{2}{6}x.
$$[/tex]
Simplify the fraction:
[tex]$$
-\frac{2}{6}x = -\frac{1}{3}x.
$$[/tex]
So, the term to add is
[tex]$$
-\frac{1}{3}x.
$$[/tex]
This corresponds to the choice:
[tex]$-\frac{1}{3} x$[/tex].
[tex]$$\frac{5}{6}x - 4$$[/tex]
and we want to determine the term that, when added, makes it equivalent to
[tex]$$\frac{1}{2}x - 4.$$[/tex]
Since both expressions have the same constant [tex]$-4$[/tex], the adjustment must be made only to the [tex]$x$[/tex]-term.
Let the term we add be [tex]$t$[/tex]. This means we need:
[tex]$$\frac{5}{6}x + t = \frac{1}{2}x.$$[/tex]
To find [tex]$t$[/tex], subtract [tex]$\frac{5}{6}x$[/tex] from both sides:
[tex]$$
t = \frac{1}{2}x - \frac{5}{6}x.
$$[/tex]
To subtract these fractions, express [tex]$\frac{1}{2}$[/tex] with denominator 6:
[tex]$$
\frac{1}{2} = \frac{3}{6}.
$$[/tex]
Thus, we have:
[tex]$$
t = \frac{3}{6}x - \frac{5}{6}x = \left(\frac{3-5}{6}\right)x = -\frac{2}{6}x.
$$[/tex]
Simplify the fraction:
[tex]$$
-\frac{2}{6}x = -\frac{1}{3}x.
$$[/tex]
So, the term to add is
[tex]$$
-\frac{1}{3}x.
$$[/tex]
This corresponds to the choice:
[tex]$-\frac{1}{3} x$[/tex].
