Question
In order to make the expression below equivalent to 1/2x+6, which additional operation should be included in the expression?
5/6x + 6 +____
−3/4x
negative 3 fourths x
−x
negative x
−2x
negative 2 x
3/4x
5/6x + 6 +____
−3/4x
negative 3 fourths x
−x
negative x
−2x
negative 2 x
3/4x
Answers
GPT-4o mini
To determine which additional operation should be included in the expression \( \frac{5}{6}x + 6 + \_ \) in order to make it equivalent to \( \frac{1}{2}x + 6 \), let's analyze the equation.
1. Start with the expression we want:
\[
\frac{1}{2}x + 6
\]
2. The expression we currently have is:
\[
\frac{5}{6}x + 6 + \_
\]
3. Since both expressions contain the term \( +6 \), we can focus on the \( x \) coefficients.
4. The coefficient of \( x \) in the expression \( \frac{1}{2}x + 6 \) is \( \frac{1}{2} \).
5. The coefficient of \( x \) in the expression \( \frac{5}{6}x + 6 \) is \( \frac{5}{6} \).
Now, we need to find a value that, when added to \( \frac{5}{6} \), will give us \( \frac{1}{2} \).
6. We can set up the equation:
\[
\frac{5}{6} + c = \frac{1}{2}
\]
where \( c \) is the additional amount we will add.
7. Convert \( \frac{1}{2} \) to sixths so we can easily solve:
\[
\frac{1}{2} = \frac{3}{6}
\]
8. Now we have:
\[
\frac{5}{6} + c = \frac{3}{6}
\]
9. Rearranging gives:
\[
c = \frac{3}{6} - \frac{5}{6} = -\frac{2}{6} = -\frac{1}{3}
\]
Now, we have found that \( c \) (the additional term) must be \( -\frac{1}{3} \).
However, among the choices given:
1. **−3/4x**
2. **−x**
3. **−2x**
4. **3/4x**
None of these choices equal \( -\frac{1}{3}x \).
Let's analyze each choice in terms of the impact on the coefficient of \( x \) to see which one could potentially result in a net equivalent coefficient of \( \frac{1}{2} \):
- Adding **−3/4x** would result in:
\[
\frac{5}{6}x - \frac{3}{4}x \implies \left( \frac{5}{6} - \frac{9}{12} \right) = \left( \frac{10}{12} - \frac{9}{12} \right) = \frac{1}{12}x \quad (\text{not } \frac{1}{2})
\]
- Adding **−x** results in:
\[
\frac{5}{6}x - 1x = \left( \frac{5}{6} - \frac{6}{6} \right) = -\frac{1}{6}x \quad (\text{not } \frac{1}{2})
\]
- Adding **−2x** results in:
\[
\frac{5}{6}x - 2x = \left( \frac{5}{6} - \frac{12}{6} \right) = -\frac{7}{6}x \quad (\text{not } \frac{1}{2})
\]
- Adding **3/4x** results in:
\[
\frac{5}{6}x + \frac{3}{4}x = \left( \frac{10}{12} + \frac{9}{12} \right) = \frac{19}{12}x \quad (\text{not } \frac{1}{2})
\]
Since none of these options achieve the desired \( \frac{1}{2}x \), it appears they are incorrect in providing equivalency as expected.
Based on the coefficients given to the operation of \(-\frac{1}{3}\) for \( x \), plus the terms given, it seems there is a mismatch or misunderstanding in potential operation terms you've shared.
The correct answer must be \( -\frac{1}{3} \) if given as a choice for the equation to balance to \( \frac{1}{2}x \) but in provided choices, there isn't a viable match or resolution towards attaining desired values from existing options.
1. Start with the expression we want:
\[
\frac{1}{2}x + 6
\]
2. The expression we currently have is:
\[
\frac{5}{6}x + 6 + \_
\]
3. Since both expressions contain the term \( +6 \), we can focus on the \( x \) coefficients.
4. The coefficient of \( x \) in the expression \( \frac{1}{2}x + 6 \) is \( \frac{1}{2} \).
5. The coefficient of \( x \) in the expression \( \frac{5}{6}x + 6 \) is \( \frac{5}{6} \).
Now, we need to find a value that, when added to \( \frac{5}{6} \), will give us \( \frac{1}{2} \).
6. We can set up the equation:
\[
\frac{5}{6} + c = \frac{1}{2}
\]
where \( c \) is the additional amount we will add.
7. Convert \( \frac{1}{2} \) to sixths so we can easily solve:
\[
\frac{1}{2} = \frac{3}{6}
\]
8. Now we have:
\[
\frac{5}{6} + c = \frac{3}{6}
\]
9. Rearranging gives:
\[
c = \frac{3}{6} - \frac{5}{6} = -\frac{2}{6} = -\frac{1}{3}
\]
Now, we have found that \( c \) (the additional term) must be \( -\frac{1}{3} \).
However, among the choices given:
1. **−3/4x**
2. **−x**
3. **−2x**
4. **3/4x**
None of these choices equal \( -\frac{1}{3}x \).
Let's analyze each choice in terms of the impact on the coefficient of \( x \) to see which one could potentially result in a net equivalent coefficient of \( \frac{1}{2} \):
- Adding **−3/4x** would result in:
\[
\frac{5}{6}x - \frac{3}{4}x \implies \left( \frac{5}{6} - \frac{9}{12} \right) = \left( \frac{10}{12} - \frac{9}{12} \right) = \frac{1}{12}x \quad (\text{not } \frac{1}{2})
\]
- Adding **−x** results in:
\[
\frac{5}{6}x - 1x = \left( \frac{5}{6} - \frac{6}{6} \right) = -\frac{1}{6}x \quad (\text{not } \frac{1}{2})
\]
- Adding **−2x** results in:
\[
\frac{5}{6}x - 2x = \left( \frac{5}{6} - \frac{12}{6} \right) = -\frac{7}{6}x \quad (\text{not } \frac{1}{2})
\]
- Adding **3/4x** results in:
\[
\frac{5}{6}x + \frac{3}{4}x = \left( \frac{10}{12} + \frac{9}{12} \right) = \frac{19}{12}x \quad (\text{not } \frac{1}{2})
\]
Since none of these options achieve the desired \( \frac{1}{2}x \), it appears they are incorrect in providing equivalency as expected.
Based on the coefficients given to the operation of \(-\frac{1}{3}\) for \( x \), plus the terms given, it seems there is a mismatch or misunderstanding in potential operation terms you've shared.
The correct answer must be \( -\frac{1}{3} \) if given as a choice for the equation to balance to \( \frac{1}{2}x \) but in provided choices, there isn't a viable match or resolution towards attaining desired values from existing options.