Answer :
To solve the problem of determining what term can be added to the expression [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 need to find a term that, when added to [tex]\(\frac{5}{6} x - 4\)[/tex], will result in the expression [tex]\(\frac{1}{2} x - 4\)[/tex].
2. Set Up the Equation:
Let [tex]\( t \)[/tex] be the term we need to find. Then according to the problem:
[tex]\[
\frac{5}{6} x - 4 + t = \frac{1}{2} x - 4
\][/tex]
3. Isolate the Term:
Since the terms [tex]\(-4\)[/tex] on both sides are the same, they cancel each other out. So we focus on the [tex]\(x\)[/tex] terms:
[tex]\[
\frac{5}{6} x + t = \frac{1}{2} x
\][/tex]
4. Solve for the Term [tex]\( t \)[/tex]:
To find [tex]\( t \)[/tex], subtract [tex]\(\frac{5}{6} x\)[/tex] from [tex]\(\frac{1}{2} x\)[/tex]:
[tex]\[
t = \frac{1}{2} x - \frac{5}{6} x
\][/tex]
5. Calculate the Subtraction:
First, convert [tex]\(\frac{1}{2}\)[/tex] to a fraction with a common denominator with [tex]\(\frac{5}{6}\)[/tex]. The common denominator is 6, so:
[tex]\[
\frac{1}{2} x = \frac{3}{6} x
\][/tex]
Now perform the subtraction:
[tex]\[
\frac{3}{6} x - \frac{5}{6} x = -\frac{2}{6} x = -\frac{1}{3} x
\][/tex]
Therefore, the term you need to 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], which matches one of the given choices.
1. Understand the Problem:
We need to find a term that, when added to [tex]\(\frac{5}{6} x - 4\)[/tex], will result in the expression [tex]\(\frac{1}{2} x - 4\)[/tex].
2. Set Up the Equation:
Let [tex]\( t \)[/tex] be the term we need to find. Then according to the problem:
[tex]\[
\frac{5}{6} x - 4 + t = \frac{1}{2} x - 4
\][/tex]
3. Isolate the Term:
Since the terms [tex]\(-4\)[/tex] on both sides are the same, they cancel each other out. So we focus on the [tex]\(x\)[/tex] terms:
[tex]\[
\frac{5}{6} x + t = \frac{1}{2} x
\][/tex]
4. Solve for the Term [tex]\( t \)[/tex]:
To find [tex]\( t \)[/tex], subtract [tex]\(\frac{5}{6} x\)[/tex] from [tex]\(\frac{1}{2} x\)[/tex]:
[tex]\[
t = \frac{1}{2} x - \frac{5}{6} x
\][/tex]
5. Calculate the Subtraction:
First, convert [tex]\(\frac{1}{2}\)[/tex] to a fraction with a common denominator with [tex]\(\frac{5}{6}\)[/tex]. The common denominator is 6, so:
[tex]\[
\frac{1}{2} x = \frac{3}{6} x
\][/tex]
Now perform the subtraction:
[tex]\[
\frac{3}{6} x - \frac{5}{6} x = -\frac{2}{6} x = -\frac{1}{3} x
\][/tex]
Therefore, the term you need to 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], which matches one of the given choices.
