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 :

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].