College

What term can you add to [tex]\frac{5}{6} x - 4[/tex] to make it equivalent to [tex]\frac{1}{2} x - 4[/tex]?

A. [tex]-\frac{1}{3} x[/tex]
B. [tex]-\frac{1}{3}[/tex]
C. [tex]\frac{1}{2} x[/tex]
D. [tex]\frac{1}{2}[/tex]

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].