Find the value of X: `(5)/(4) -(7)/(6)-((-2)/(3))`=X

Options

A. `(3)/(4)`
B. `(-3)/(4)`
C. `(-7)/(12)`
D. `(7)/(12)`

Answer

To find the value of \( X \) in the equation

\[ \frac{5}{4} - \frac{7}{6} - \left(-\frac{2}{3}\right) = X, \]

we will follow these steps:

### Step 1: Simplify the equation
First, we can rewrite the equation by converting the double negative into a positive:

\[ X = \frac{5}{4} - \frac{7}{6} + \frac{2}{3}. \]

### Step 2: Find the LCM of the denominators
The denominators are 4, 6, and 3. We need to find the least common multiple (LCM) of these numbers.

- The prime factorization of 4 is \( 2^2 \).
- The prime factorization of 6 is \( 2^1 \times 3^1 \).
- The prime factorization of 3 is \( 3^1 \).

The LCM will take the highest power of each prime:

- For 2, the highest power is \( 2^2 \) (from 4).
- For 3, the highest power is \( 3^1 \) (from both 6 and 3).

Thus, the LCM is:

\[ LCM = 2^2 \times 3^1 = 4 \times 3 = 12. \]

### Step 3: Convert each fraction to have the common denominator
Now we convert each fraction to have a denominator of 12:

- For \( \frac{5}{4} \):
\[ \frac{5}{4} = \frac{5 \times 3}{4 \times 3} = \frac{15}{12}. \]

- For \( \frac{7}{6} \):
\[ \frac{7}{6} = \frac{7 \times 2}{6 \times 2} = \frac{14}{12}. \]

- For \( \frac{2}{3} \):
\[ \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}. \]

### Step 4: Substitute back into the equation
Now substitute these values back into the equation:

\[ X = \frac{15}{12} - \frac{14}{12} + \frac{8}{12}. \]

### Step 5: Combine the fractions
Now we can combine the fractions:

\[ X = \frac{15 - 14 + 8}{12} = \frac{9}{12}. \]

### Step 6: Simplify the fraction
Finally, simplify \( \frac{9}{12} \):

\[ \frac{9}{12} = \frac{3}{4}. \]

Thus, the value of \( X \) is

\[ X = \frac{3}{4}. \]

### Summary of Steps:
1. Rewrite the equation to eliminate the double negative.
2. Find the LCM of the denominators.
3. Convert each fraction to have the common denominator.
4. Substitute back into the equation.
5. Combine the fractions.
6. Simplify the final fraction.