To verify the equation
[tex]\[
\frac{-3}{4}+\left(\frac{1}{2}+\left(\frac{-7}{6}\right)\right)=\left(\frac{3}{4}+\frac{1}{2}\right)+\frac{-7}{6},
\][/tex]
we will evaluate each side of the equation separately and then compare the results.
First, let's evaluate the left-hand side (LHS):
[tex]\[
\text{LHS} = \frac{-3}{4} + \left(\frac{1}{2} + \frac{-7}{6}\right)
\][/tex]
Step 1: Calculate the value inside the parentheses:
[tex]\[
\frac{1}{2} + \frac{-7}{6}
\][/tex]
To add these fractions, we need a common denominator. The least common denominator of 2 and 6 is 6, so we convert each fraction:
[tex]\[
\frac{1}{2} = \frac{3}{6}, \quad \frac{-7}{6} = \frac{-7}{6}
\][/tex]
Now, we add the fractions:
[tex]\[
\frac{3}{6} + \frac{-7}{6} = \frac{3 - 7}{6} = \frac{-4}{6} = \frac{-2}{3}
\][/tex]
Step 2: Now, add this result to \(\frac{-3}{4}\):
[tex]\[
\frac{-3}{4} + \frac{-2}{3}
\][/tex]
Again, we need a common denominator to add these fractions. The least common denominator of 4 and 3 is 12, so we convert each fraction:
[tex]\[
\frac{-3}{4} = \frac{-9}{12}, \quad \frac{-2}{3} = \frac{-8}{12}
\][/tex]
Now, we add the fractions:
[tex]\[
\frac{-9}{12} + \frac{-8}{12} = \frac{-9 - 8}{12} = \frac{-17}{12}
\][/tex]
So, the left-hand side simplifies to:
[tex]\[
\text{LHS} = \frac{-17}{12} = -1.4166666666666667
\][/tex]
Next, let's evaluate the right-hand side (RHS):
[tex]\[
\text{RHS} = \left(\frac{3}{4} + \frac{1}{2}\right) + \frac{-7}{6}
\][/tex]
Step 1: Calculate the value inside the parentheses:
[tex]\[
\frac{3}{4} + \frac{1}{2}
\][/tex]
Again, we need a common denominator. The least common denominator of 4 and 2 is 4, so we convert each fraction:
[tex]\[
\frac{3}{4} = \frac{3}{4}, \quad \frac{1}{2} = \frac{2}{4}
\][/tex]
Now, we add the fractions:
[tex]\[
\frac{3}{4} + \frac{2}{4} = \frac{3 + 2}{4} = \frac{5}{4}
\][/tex]
Step 2: Now, add this result to \(\frac{-7}{6}\):
[tex]\[
\frac{5}{4} + \frac{-7}{6}
\][/tex]
We need a common denominator to add these fractions. The least common denominator of 4 and 6 is 12, so we convert each fraction:
[tex]\[
\frac{5}{4} = \frac{15}{12}, \quad \frac{-7}{6} = \frac{-14}{12}
\][/tex]
Now, we add the fractions:
[tex]\[
\frac{15}{12} + \frac{-14}{12} = \frac{15 - 14}{12} = \frac{1}{12}
\][/tex]
So, the right-hand side simplifies to:
[tex]\[
\text{RHS} = \frac{1}{12} = 0.08333333333333326
\][/tex]
Finally, we compare the two sides. The left-hand side is \(-1.4166666666666667\) and the right-hand side is \(0.08333333333333326\).
Since \(\text{LHS} \ne \text{RHS}\), the original equation
[tex]\[
\frac{-3}{4}+\left(\frac{1}{2}+\left(\frac{-7}{6}\right)\right)\neq\left(\frac{3}{4}+\frac{1}{2}\right)+\frac{-7}{6}
\][/tex]
is not valid.
