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.

Converting both fractions:

\[ \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \]
\[ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \]

Now, we can subtract:

\[ x - y = \frac{8}{12} - \frac{9}{12} = \frac{8 - 9}{12} = \frac{-1}{12} \]

### Step 2: Calculate \( (x - y)^{-1} \)

Now we find the reciprocal:

\[ (x - y)^{-1} = \left(\frac{-1}{12}\right)^{-1} = -12 \]

### Step 3: Calculate \( x^{-1} \) and \( y^{-1} \)

Next, we calculate the reciprocals of \( x \) and \( y \):

\[ x^{-1} = \left(\frac{2}{3}\right)^{-1} = \frac{3}{2} \]
\[ y^{-1} = \left(\frac{3}{4}\right)^{-1} = \frac{4}{3} \]

### Step 4: Calculate \( x^{-1} - y^{-1} \)

Now we subtract \( y^{-1} \) from \( x^{-1} \):

\[ x^{-1} - y^{-1} = \frac{3}{2} - \frac{4}{3} \]

Again, we need a common denominator. The LCM of 2 and 3 is 6.

Converting both fractions:

\[ \frac{3}{2} = \frac{3 \times 3}{2 \times 3} = \frac{9}{6} \]
\[ \frac{4}{3} = \frac{4 \times 2}{3 \times 2} = \frac{8}{6} \]

Now we can 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 can combine the results from Step 2 and Step 4:

\[ (x - y)^{-1} + (x^{-1} - y^{-1}) = -12 + \frac{1}{6} \]

To add these, we convert \(-12\) to a fraction with a denominator of 6:

\[ -12 = \frac{-72}{6} \]

Now we can add:

\[ \frac{-72}{6} + \frac{1}{6} = \frac{-72 + 1}{6} = \frac{-71}{6} \]

### Final Answer

Thus, the value of the expression is:

\[ \boxed{\frac{-71}{6}} \]

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