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 solve this problem, we need to determine what term should be added to the expression [tex]\(\frac{5}{6} x - 4\)[/tex] to make it equivalent to [tex]\(\frac{1}{2} x - 4\)[/tex].

### Step-by-Step Solution:

1. Compare the Expressions:

We want [tex]\(\frac{5}{6} x - 4 + \text{(some term)} = \frac{1}{2} x - 4\)[/tex].

2. Match the Constant Terms:

Notice that the constant terms (-4) on both sides are already the same. So, we only need to focus on matching the coefficients of [tex]\(x\)[/tex].

3. Set Up an Equation for the Coefficients of [tex]\(x\)[/tex]:

Since the constant terms are equal, focus on the [tex]\(x\)[/tex] terms:
[tex]\[
\frac{5}{6} x + \text{(some term)} = \frac{1}{2} x
\][/tex]

This implies:
[tex]\[
\text{some term} = \frac{1}{2} x - \frac{5}{6} x
\][/tex]

4. Calculate the Term:

Subtract the coefficients of [tex]\(x\)[/tex]:
[tex]\[
\frac{1}{2} - \frac{5}{6}
\][/tex]

To subtract these fractions, find a common denominator, which is 6 in this case:
[tex]\[
\frac{1}{2} = \frac{3}{6}
\][/tex]
Then:
[tex]\[
\frac{3}{6} - \frac{5}{6} = -\frac{2}{6}
\][/tex]
Simplifying [tex]\(-\frac{2}{6}\)[/tex], we get:
[tex]\[
-\frac{1}{3}
\][/tex]

5. Conclusion:

The term you need to add is [tex]\(-\frac{1}{3} x\)[/tex]. Therefore, the answer is:

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

This matches the option provided: [tex]\(-\frac{1}{3} x\)[/tex]. Thus, this is the term that will make the two expressions equivalent.