Certainly! Let's verify the given equation step-by-step.
### Left Side of the Equation
First, we need to evaluate the left side of the equation:
[tex]\[
\frac{-3}{4} + \left[ \frac{1}{2} + \left( \frac{-1}{6} \right) \right]
\][/tex]
1. Calculate inside the innermost brackets:
[tex]\[
\left( \frac{1}{2} + \left( \frac{-1}{6} \right) \right) = \frac{1}{2} - \frac{1}{6}
\][/tex]
To subtract these fractions, we need a common denominator. The common denominator for 2 and 6 is 6.
[tex]\[
\frac{1}{2} = \frac{3}{6}
\][/tex]
So,
[tex]\[
\frac{3}{6} - \frac{1}{6} = \frac{2}{6} = \frac{1}{3}
\][/tex]
2. Add the result to \(\frac{-3}{4}\):
[tex]\[
\frac{-3}{4} + \frac{1}{3}
\][/tex]
Again, find a common denominator for 4 and 3, which is 12.
[tex]\[
\frac{-3}{4} = \frac{-9}{12} \quad \text{and} \quad \frac{1}{3} = \frac{4}{12}
\][/tex]
Thus,
[tex]\[
\frac{-9}{12} + \frac{4}{12} = \frac{-5}{12}
\][/tex]
So, the left side of the equation simplifies to:
[tex]\[
\frac{-5}{12}
\][/tex]
### Right Side of the Equation
Next, let's evaluate the right side of the equation:
[tex]\[
\left[ \frac{3}{4} + \frac{1}{2} \right] + \frac{-7}{6}
\][/tex]
1. Calculate inside the brackets:
[tex]\[
\left( \frac{3}{4} + \frac{1}{2} \right)
\][/tex]
Convert \(\frac{1}{2}\) to have a common denominator with \(\frac{3}{4}\), which is 4.
[tex]\[
\frac{1}{2} = \frac{2}{4}
\][/tex]
So,
[tex]\[
\frac{3}{4} + \frac{2}{4} = \frac{5}{4}
\][/tex]
2. Add the result to \(\frac{-7}{6}\):
[tex]\[
\frac{5}{4} + \frac{-7}{6}
\][/tex]
Find a common denominator for 4 and 6, which is 12.
[tex]\[
\frac{5}{4} = \frac{15}{12} \quad \text{and} \quad \frac{-7}{6} = \frac{-14}{12}
\][/tex]
Thus,
[tex]\[
\frac{15}{12} + \frac{-14}{12} = \frac{1}{12}
\][/tex]
So, the right side of the equation simplifies to:
[tex]\[
\frac{1}{12}
\][/tex]
### Compare Left and Right Sides
Finally, we compare the simplified expressions of both sides:
[tex]\[
\frac{-5}{12} \quad \text{(left side)} \quad \text{and} \quad \frac{1}{12} \quad \text{(right side)}
\][/tex]
Clearly, \(\frac{-5}{12}\) does not equal \(\frac{1}{12}\).
### Conclusion
The two sides of the equation are not equal:
[tex]\[
\frac{-3}{4} + \left[ \frac{1}{2} + \left( \frac{-1}{6} \right) \right] \neq \left[ \frac{3}{4} + \frac{1}{2} \right] + \frac{-7}{6}
\][/tex]
The verification shows that the left-hand side is [tex]\(\frac{-5}{12}\)[/tex] and the right-hand side is [tex]\(\frac{1}{12}\)[/tex], proving that the equation is false.
