Sure, let's solve the given expression step-by-step.
The given expression is:
[tex]\[ 1\frac{1}{2} \div \left( 3\frac{1}{3} + 4\frac{1}{5} - 6\frac{1}{2} \right) \][/tex]
First, convert all the mixed fractions to improper fractions.
1. Convert \(1\frac{1}{2}\):
[tex]\[ 1 \frac{1}{2} = \frac{3}{2} = 1.5 \][/tex]
2. Convert \(3\frac{1}{3}\):
[tex]\[ 3 \frac{1}{3} = \frac{10}{3} \approx 3.3333333333333335 \][/tex]
3. Convert \(4\frac{1}{5}\):
[tex]\[ 4 \frac{1}{5} = \frac{21}{5} = 4.2 \][/tex]
4. Convert \(6\frac{1}{2}\):
[tex]\[ 6 \frac{1}{2} = \frac{13}{2} = 6.5 \][/tex]
Now, substitute these values into the expression inside the parentheses:
[tex]\[ 3\frac{1}{3} + 4\frac{1}{5} - 6\frac{1}{2} \approx 3.3333333333333335 + 4.2 - 6.5 \][/tex]
Calculate the sum and the difference inside the parentheses:
[tex]\[ 3.3333333333333335 + 4.2 - 6.5 \approx 1.0333333333333332\ ]
Now the expression becomes:
\[ 1\frac{1}{2} \div 1.0333333333333332 \][/tex]
Finally, perform the division:
[tex]\[ 1.5 \div 1.0333333333333332 \approx 1.4516129032258067 \][/tex]
So, the value of the given expression is:
[tex]\[ \boxed{1.4516129032258067} \][/tex]
