High School

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 determine which term can be added to [tex]\(\frac{5}{6}x - 4\)[/tex] to make it equivalent to [tex]\(\frac{1}{2}x - 4\)[/tex], follow these steps:

1. Understand the Problem:
We are looking to make the two expressions equivalent:
[tex]\[
\frac{5}{6}x - 4 + \text{term} = \frac{1}{2}x - 4
\][/tex]

2. Remove the Common Part:
Both expressions have [tex]\(-4\)[/tex], so you can ignore it for now to simplify the problem:
[tex]\[
\frac{5}{6}x + \text{term} = \frac{1}{2}x
\][/tex]

3. Solve for the Missing Term:
To find the term, subtract [tex]\(\frac{5}{6}x\)[/tex] from both sides:
[tex]\[
\text{term} = \frac{1}{2}x - \frac{5}{6}x
\][/tex]

4. Calculate the Difference:
Convert [tex]\(\frac{1}{2}x\)[/tex] to have a common denominator with [tex]\(\frac{5}{6}x\)[/tex]. The common denominator for 2 and 6 is 6. So, [tex]\(\frac{1}{2}x\)[/tex] becomes [tex]\(\frac{3}{6}x\)[/tex].

Now, subtract:
[tex]\[
\text{term} = \frac{3}{6}x - \frac{5}{6}x = -\frac{2}{6}x
\][/tex]

5. Simplify the Term:
Simplify [tex]\(-\frac{2}{6}x\)[/tex]:
[tex]\[
\text{term} = -\frac{1}{3}x
\][/tex]

So, the term you can add to [tex]\(\frac{5}{6}x - 4\)[/tex] to make it equivalent to [tex]\(\frac{1}{2}x - 4\)[/tex] is [tex]\(-\frac{1}{3}x\)[/tex]. Therefore, the correct answer is:

[tex]\(-\frac{1}{3}x\)[/tex]