If `x = (2)/(3)` and `y = (3)/(4)`, then a rational number `(x-y)^(-1)+(x^(-1)- y^(-1))` is equal to

Options

A. `(2)/(3)`
B. `(-71)/(6)`
C. `-(71)/(12)`
D. `(1)/(3)`

Answer

To solve the expression \( (x - y)^{-1} + (x^{-1} - y^{-1}) \) given \( x = \frac{2}{3} \) and \( y = \frac{3}{4} \), we will follow these steps:

### Step 1: Calculate \( x - y \)
\[ x - y = \frac{2}{3} - \frac{3}{4} \]
To perform this subtraction, we need a common denominator. The least common multiple (LCM) of 3 and 4 is 12.

Convert each fraction:
\[ \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \]
\[ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \]

Now, subtract:
\[ x - y = \frac{8}{12} - \frac{9}{12} = \frac{8 - 9}{12} = \frac{-1}{12} \]

### Step 2: Calculate \( (x - y)^{-1} \)
\[ (x - y)^{-1} = \left(\frac{-1}{12}\right)^{-1} = -12 \]

### Step 3: Calculate \( x^{-1} \) and \( y^{-1} \)
\[ x^{-1} = \left(\frac{2}{3}\right)^{-1} = \frac{3}{2} \]
\[ y^{-1} = \left(\frac{3}{4}\right)^{-1} = \frac{4}{3} \]

### Step 4: Calculate \( x^{-1} - y^{-1} \)
\[ x^{-1} - y^{-1} = \frac{3}{2} - \frac{4}{3} \]
Again, we need a common denominator. The LCM of 2 and 3 is 6.

Convert each fraction:
\[ \frac{3}{2} = \frac{3 \times 3}{2 \times 3} = \frac{9}{6} \]
\[ \frac{4}{3} = \frac{4 \times 2}{3 \times 2} = \frac{8}{6} \]

Now, subtract:
\[ x^{-1} - y^{-1} = \frac{9}{6} - \frac{8}{6} = \frac{9 - 8}{6} = \frac{1}{6} \]

### Step 5: Combine the results
Now we combine the results from Step 2 and Step 4:
\[ (x - y)^{-1} + (x^{-1} - y^{-1}) = -12 + \frac{1}{6} \]
To add these, convert \(-12\) into a fraction with a denominator of 6:
\[ -12 = \frac{-72}{6} \]
Now, add:
\[ \frac{-72}{6} + \frac{1}{6} = \frac{-72 + 1}{6} = \frac{-71}{6} \]

### Final Answer
Thus, the value of \( (x - y)^{-1} + (x^{-1} - y^{-1}) \) is:
\[ \frac{-71}{6} \]