To verify the given equation, we need to simplify both the left-hand side (LHS) and the right-hand side (RHS) and see if they are equal.
First, let’s simplify the LHS:
LHS:
[tex]\[
\frac{-3}{4} + \left[\frac{1}{2} + \left(\frac{-7}{6}\right)\right]
\][/tex]
1. Calculate \(\frac{1}{2} + \left(\frac{-7}{6}\right)\):
[tex]\[
\frac{1}{2} = \frac{3}{6}
\][/tex]
So,
[tex]\[
\frac{3}{6} + \frac{-7}{6} = \frac{3 - 7}{6} = \frac{-4}{6} = \frac{-2}{3}
\][/tex]
2. Now add \(\frac{-3}{4}\):
[tex]\[
\frac{-3}{4} + \frac{-2}{3}
\][/tex]
Find a common denominator, which is 12:
[tex]\[
\frac{-3}{4} = \frac{-9}{12} \quad \text{and} \quad \frac{-2}{3} = \frac{-8}{12}
\][/tex]
So,
[tex]\[
\frac{-9}{12} + \frac{-8}{12} = \frac{-17}{12}
\][/tex]
Next, simplify the RHS:
RHS:
[tex]\[
\left[\frac{3}{4}+\frac{1}{2}\right]+\frac{-7}{6}
\][/tex]
1. Calculate \(\frac{3}{4} + \frac{1}{2}\):
[tex]\[
\frac{1}{2} = \frac{2}{4}
\][/tex]
So,
[tex]\[
\frac{3}{4} + \frac{2}{4} = \frac{5}{4}
\][/tex]
2. Now add \(\frac{-7}{6}\):
[tex]\[
\frac{5}{4} + \frac{-7}{6}
\][/tex]
Find a common denominator, which is 12:
[tex]\[
\frac{5}{4} = \frac{15}{12} \quad \text{and} \quad \frac{-7}{6} = \frac{-14}{12}
\][/tex]
So,
[tex]\[
\frac{15}{12} + \frac{-14}{12} = \frac{15 - 14}{12} = \frac{1}{12}
\][/tex]
So now we compare both sides:
LHS:
[tex]\[
\frac{-17}{12} \approx -1.4166666666666667
\][/tex]
RHS:
[tex]\[
\frac{1}{12} \approx 0.08333333333333326
\][/tex]
Clearly:
[tex]\[
\frac{-17}{12} \neq \frac{1}{12}
\][/tex]
Therefore, the left-hand side is not equal to the right-hand side.
